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Causes of Death in Dialysis Patients: Racial and

Gender Differences1
Wendy E. Bloembergen,2 Friedrich K. Port, Elizabeth A. Mauger, and Robert A. Wolfe

WE. Bloembergen, F.K. Port, Department of Internal

P atients
Reports have
for ESRD
a high
the risk
of mortal-
Medicine, School of Medicine. University of Michigan, ity among patients treated for ESRD is higher for
Ann Arbor. Ml males than for females and for whites than for blacks.
F.K. Port, Department of Epidemiology, School of Pub- The U.S. Renal Data System (USRDS) has shown that
ic Health, University of Michigan. Ann Arbor, Ml 1 -yr survival rates for ESRD patients incident during
the 1 980s are typically two to three percentage points
E.A. Mauger, PA. Wolfe, Department of Biostatistics,
higher for females than for males when adjusted for
School of Public Health, University of Michigan. Ann
age, race, and primary cause of ESRD ( 1 ). In addition,
Arbor, Ml
it has also shown that, although young black ESRD
(J. Am. Soc. Nephrol. 1994; 5:1231-1242) patients have a higher risk of death than young white
ESRD patients, aggregating all ages, blacks have a
lower risk of death than whites (2). Likewise, in a
ABSTRACT study of all registered Canadian ESRD patients mci-
dent between 1981 and 1986, higher death rates (DR)
The risk of death in the dialysis population is high and
were found among males and white ESRD patients
has previously been shown to be accentuated in
when adjusted for age, gender on race, blood type,
male (versus female) and white (versus black) sub- primary diagnosis, transplantation, and modality of
groups. To better understand the difference in mortal- first treatment (3).
ity among these subgroups, the causes of death These gender and racial differences in mortality
between males and females as well as between have also been reported in studies restricted to the
whites and
blacks adjusting for age, cause of ESRD dialysis population in which transplant recipients are
(diabetic versus nondiabetic), dialysis modality, and excluded. A higher risk of mortality in males and
time on dialysis (< 1 yr versus > i yr) were compared, whites was found in an analysis of all Medicare-
with national data obtained from the U.S. Renal Data treated U.S. hemodialysis (HD) patients incident in
1977 (4). More recent studies have also shown these
System. A total of 42,372 deaths occurring over
gender and/or racial differences but less consistently
170,700 patient years at risk were analyzed. Males
(5-7). Studies performed on data from HD and penito-
had a 22% higher risk of death than females (P <
neal dialysis (PD) patients in Michigan (5) and HD
0.001), attributable to a higher risk of death due to patients in Georgia (6) confirmed the racial but not the
acute myocardial infarction (relative death rate ratio gender difference in DR. In the USRDS Special Study
(RR) = 1.48; P = 0.001), all other cardiac causes (RR = of Case Mix Severity, the gender difference was con-
1 .3; P = 0.001), and malignancy (PR = 1.59; P < 0.00 1). firmed among HD patients; however, the higher risk of
Whites had a 29% higher risk of death than blacks (P mortality for white compared with black patients was
< 0.001), accounted for by an increased risk of death not statistically significant (7).
due to acute myocardial infarction (PR = 1.34), all Cause of death statistics are of vital importance in
other cardiac causes (PR = 1.30), wIthdrawal from medical epidemiologic research and serve as a source
of new hypotheses that can be explored by in-depth
dialysis (RR = 2.72) (all P < 0.001), and infection (PR =
epidemiologic and clinical research. Few studies have
1 .09; P = 0.005). This analysis expands the knowledge
compared the causes of death in ESRD patients by
and understanding of the excess mortalily seen In gender or race. Those that have are somewhat limited
male and white subgroups, which is a necessary step in that they are from single programs (8) or deal with
in designing strategies to reduce the high mortality in only one on few specific causes of death (9-13). Fun-
dialysis patients. thermone, previous studies have generally compared
the percentage distribution of causes of death among
Key Words: ESRD. hemodialysis, peritoneal dialysis, mortalily, patients who died and thus have been primarily de-
death rates scniptive in nature. In this type of analysis. an ob-
served excess of one cause of death may represent a
true risk but may also merely represent a deficit of
1 Received September 15, 1994. Accepted May 19. 1994. deaths due to another cause. The USRDS has pub-
2 correspondence to Dr. WE. Bloembergen. Kidney Epidemiology and coal lished cause-specific mortality rates; however, these
Center, 315 West Huron, Suite 240. Ann Arbor, MI 48103.
have not been adjusted for important demographic
1046.6673/0505-1231 $030010
Journal of the American Society of Nephrology
covariates non analyzed to determine comparative
copyght © 1994 by the American SocIety of Nephrology risks between subgroups by race and gender. The

Journal of the American Society of Nephrology 1231

Causes of Death in Dialysis Patients

purpose of this study therefore was to confirm the Data Analysis

previously reported increased risks of mortality
among whites and males and to compare actual DR DR were compared between females and males as well as
between blacks and whites for each of the six collapsed
between race and gender subgroups for major causes
known causes of death (acute MI, “other cardiac causes,”
of death in order to achieve a better understanding of
cerebrovascular disease, infection, malignancy, and with-
the excess mortality in gender- and race-specific sub-
drawal from dialysis). Crude DR. computed as the aggregate
groups. number ofdeaths divided by the aggregate number of patient
years at risk for the three cohorts, are reported per 100
patient years at risk.
METHODS Poisson regression was used for each cause of death to
compare the DR of various subgroups while adjusting for all
Source of Data and Study Population other demographic characteristics. All comparisons were
adjusted for age, gender. race (white, black, or other), cause
Data were obtained from the USRDS. The USRDS collects of ESRD (diabetic nephropathy versus all others), and dialy-
demographic and clinical information on patients who sur- sis modality (HD or PD). It was speculated that differences in
vive at least 90 days on renal replacement therapy and specific causes of death may exist between patients who die
qualify for Medicare (approximately 93% of the total treated soon after the initiation ofESRD treatment and those who die
ESRD population). Patients receiving continuous ambula- after a longer duration of dialysis therapy. Therefore, pa-
tory peritoneal dialysis/continuous cycles-assisted perito- tients who started ESRD therapy less than or equal to 1 yr
neal dialysis (PD) or HD were included. Data as received from before each January 1 study start date were classified as
the USRDS were obtained from three cohorts of dialysis “incident” and the remainder were classified as “noninci-
patients prevalent on January 1 of 1987, 1988, and 1989, dent.” Patients who were new (‘incident”) in 1 yr were clas-
each with 1 yr of follow-up. sified as “nonincident” in subsequent years. Comparisons
Patients starting dialysis less than 3 months before Janu- were also adjusted for this characteristic. Age on January 1
ary 1 were excluded for that year because data were poten- of each year was entered by 5-yr age categories, as obtained
tially incomplete for these patients. Patients who had a from USRDS. All other covariates were entered into the model
previous renal transplant were excluded to enhance the as dichotomous variables. DR comparisons are presented as
homogeneity of the study population. Patients transplanted a relative risk (RR). which is the ratio of adjusted death rates
during the year were censored on the day of transplantation. among the two subgroups being compared, e.g. , males com-
Switches in dialytic modality during the 365 days of fol- pared with females or blacks compared with whites.
low-up were not considered. Additional analyses assessed if differences between sub-
Each cohort was monitored for one calendar year for groups in the cause-specific DR varied by other demographic
deaths, cause ofdeath, and days at risk ofdeath. Time at risk characteristics by including interaction terms in the regres-
of death was summed from the beginning of the year to the sion model. Ifan interaction was statistically significant (e.g.,
date of death, to the date of transplant, or to the end of the interaction of gender by modality), an RR was computed for
year for each patient. Patients surviving until the end of the each subgroup (e.g. , male/female IM/FI PR for PD and HD
year contributed 1 yr ofrisk, whereas patients dying midyear separately). P values
were also obtained to determine if each
contributed #{189}
yr at risk on average. A patient prevalent at of the two individual PR were significantly different from 1.0.
the beginning of more than one calendar year contributed If only one subgroup (e.g. , HD) had an RR significantly
data to more than one cohort. Patient follow-up was charac- different from 1 . the excess risk was said to be present only in
terized by age, race, gender, dialysis modality, cause of that subgroup. If, however, both RR were significantly differ-
ESRD, and time on dialysis at the beginning of each cohort ent from 1 , the risk (e.g. , gender) was said to be accentuated
year, better reflecting the effect ofcovariates that change in a in the subgroup with the greater RR. Statistical analyses
given patient from one cohort year to another (age, modality, were performed with the PROC LOGISTIC procedure of SAS
and time on dialysis). The total number of patient deaths and v6.01 (Cary, NC).
years at risk for the three cohorts were aggregated to enhance For comparison of overall DR, all deaths were included
the stability of the estimated DR. (N = 42,372). In 18.0% of the deaths during 1987 to 1989,
For each death occurring in the U.S. ESRD population. the the cause ofdeath was either unknown or missing. To ensure
Health Care Financing Administration (HCFA) requires the that a difference in a DR due to a specific cause was not
primary cause of death to be reported by the patient’s renal simply due to a difference in the proportion ofpatients within
physician. This occurs by means of a Death Notification a subgroup with cause of death missing/unknown, both the
Form (HCFA-2746), which lists 22 cause of death categories number of deaths (N = 7.653) and the number of patient
(e.g. . acute myocardial infarction IMII, sepsis, etc.) The years that these patients contributed (3 1 90 1 patient years)
USRDS data base includes these data as well as a “missing” for the missing/unknown category were excluded for the
cause of death category for those patients who, by means of comparison of individual causes of death. This assumes that
the Social Security System and/or hospital discharge the distribution of cause of death in those patients who have
records. are known to have died but for whom no Death an unknown cause of death or missing data is similar to the
Notification Form was received. For the purpose of this distributions noted in the patients with cause of death
analysis, these 23 cause of death categories were collapsed reported.
into 8 categories: acute myocardial infarction, “other cardiac To account for the multiple comparisons performed, we
causes” (all cardiac causes other than acute MI), cerebrovas- applied the Bonferroni procedure to this analysis of six
cular disease, infection, malignancy, withdrawal from dialy- causes of death, requiring a P value of less than 0.008 (0.05
sis, other, and unknown/missing. Because revisions to the divided by 6) to conclude that a significant difference existed.
cause of death categories on the Death Notification Form Similarly. to conclude that there was a significant interaction
were made in 1990, this study uses data only through 1989. with another covariate, a P value less than 0.01 (0.05/5) was

1232 Volume 5 - Number 5 - 1994

Bloembergen et al

required because five covariates were tested. We chose to Death Rate (per 100 patient years)
label an RR value of 0.95 to 1 .05 or a P value of greater or 40
equal to 0.0 1 as not significantly different from 1 in the
comparisons of gender (or race) risk within subgroups (e.g.,
PD and HD). Because the Bonferroni correction provides a 30
conservative statistical criterion, true differences may be
missed. The data are therefore presented with the level of
confidence noted. Items that are significant with the Bonfer- 19.7
roni correction are indicated.

A total of 42,372 deaths occurred over 170,700 p’O.OOl’

patient years at risk in this study population of

chronic dialysis patients. The study population was
60% white, 36% black, and 4% other races. Almost
50% were females. Other demographic features are Female Male
shown in Table 1.
Figure 1 . Adjusted all-cause DR for females and males. These
total DR are for dialysis patients with the average character-
Gender istics of the total study population. PR, DR ratIo (M/F), ad-
Of deaths occurring between 1987 and 1989, justed for age, race, diabetes, dialysis modality, and dura-
22,26 1 occurred in males and 20, 1 1 1 deaths occurred Hon of dialysis (< or > 1 yr).
in females. DR pen 100 patIent years at risk for all
causes and for each cause individually were compared Death Rate (per 100 patient years)
10 -- --- --- -
between males and females with statistical adjust-
ment for other factors. Overall, for all causes, the DR
- Fsmales Males
for males was 22% hIgher than that for females (RR = 8
1 .22; P < 0.00 1 ), adjusted for age, race, diabetic
versus nondiabetic ESRD, dialysis modality, and “in-
cident” versus “nonincident.” Figure 1 shows DR for 6

males (24. 1 / 100 patIent years) compared with fe-

males (19.7/ 100 patient years) for patients with the
average characteristics of the total study population.
Figure 2 shows the DR for each individual cause of
death. DR due to “other cardiac causes” (all cardiac 2 - k---- RR-1.59
RR-1.48 RR#{149}1.30 #{149}--- RR1.08 p’O.OOl -
causes other than acute MI) were the highest, more
p0.001 p-0.001 p.0.01
- ,
than twofold greater than the next leading cause of
death, which was infection for females and acute
0 Li i
MI Oth.Cardiac CVD Infection Malig Withdrawal
myocardial infarction for males. Withdrawal from di-
alysis, cerebrovascular disease, and malignancy con- Figure 2. DR for individual causes of death and comparisons
stituted the next highest death rates in decreasing between females and males. DR are for patients with aver-
order. This figure also shows the M/F RR for dialysis
age characteristics of the study population. RR, DR ratio
patients. In particular, males have a higher DR due to
(M/F), adjusted for age, race, diabetes, dialysis modality,
and duration of dialysis (< or > 1 yr). 0th. Cardiac, all
acute myocardial infarction (RR = 1 .48; P = 0.001),
cardiac causes other than Ml; CVD, cerebrovascular dis-
ease; Malig, malignancy; Withdrawal, withdrawal from dial-
TABLE 1 Demographic
. characteristics of study ysis. P < 0.05. P < 0.008 (significant at the 0.05 level with
population Bonferroni correction).

%Whtte 60.4 “other cardiac causes” (RR = 1 .30; P = 0.00 1 ), and
% Black 35.8 malignancy (RR = 1 .59; P < 0.00 1) than females. The
%Other 3.8 DR for males and females on dialysis are not signifi-
Gender cantly different (P > 0.008) for infection (RR = 1.08;
% Female 49.8 P = 0.0 1 ), cerebrovascular disease (PR = 0.93; P =
Mean Age, yr (SD) 60.2 (9.3) 0.05), and withdrawal from dialysis (RR = 1 .03; P
Cause of ESRD
0.37) when the Bonferroni correction is applied.
%Dlabetlc Nephropathy 25.0
Further subanalyses were done to determine if the
observed M/F differences for total deaths on for each
% Hemodialysis 87.1
individual cause of death varied by race, cause of
Time on Dialysis
% Incident (< 1 yr) ESRD (diabetes versus other causes), dialysis modal-
ity, on length of time on dialysis (“incident” = less than

Journal of the American Society of Nephrology 1233

Causes of Death in Dialysis Patients

TABLE 2. Adjusted M/F RR of death for each cause of death by race, cause, modality, and time on dialysis

Race Cause of ESRD Modality Time on Dialysis

Cause of Death Overall
Black White Nondiabetes Diabetes HD PD Incident Prevalent

Total 1.22#{176} 1#{149}23a,b 1.2#{176} 1.23#{176} 1.20#{176} 1.25a 1.ooa 1.19 1.24a
Ml 1.48#{176} 1.40#{176} 1.50a 1.56 1.35 1.52#{176} 1.33a 1.40#{176} 1.52a
Other Cardiac 1.30#{176} 1.22#{176} 1.32#{176} 1.30#{176} 1.24#{176} 1.30#{176} 1.24#{176} 1.26#{176} 1.31#{176}
Cerebrovascular Disease 0.93 0.93 0.92 0.92 0.90 0.96 0.73c 0.84 0.95
Withdrawal 1.03 1.03 1.03 0.98 1.16 1.06 0.85 1.04 1.03
Malignancy 1 .59#{176} 1.52#{176} 1.62#{176} 1.60 1.51 1 .61C 1.42 1.49#{176} 1.64#{176}
Infection 1.08 1.19c 1.01 1.04 1.15c 1.lla 0.90 1.00 1.lOc
0 Different from 1.00; p < 0.001.
b Boldface areas represent PP that are significantly different from each other (interaction is present) with Bonferroni correction (P < 0.01).
C Different from 1.00; P 0.01.

or equal to 1 yr on “nonincident” = greaten than 1 yr). Acute Myocardial Infarction. The M/F RR due to
The results for race, cause, modality, and time on acute myocardial infarction was significantly accentu-
dialysis are summarized in Table 2. Interactions ated (P = 0.005) in patients with nondiabetic causes of
found to be significant are discussed subsequently. ESRD compared with patients with diabetes (intenac-
Deaths due to All Causes. Although the “all-cause” tion was present). The M/F RR was 1 .56 in nondia-
DR was higher for males compared with females, betics and 1 .35 in diabetics (Figure 4). In addition, the
further analysis showed that this difference was sig- M/F RR for acute MI increases with age (P = 0.003). At
nificantly greater among patients treated with HD ages older than 65, there is a significant difference in
than for patients treated with PD (P < 0.00 1 ). The DR by gender (P < 0.0 1 ). At ages younger than 47,
increased DR for males compared with females in there is no significant difference in DR by gender (P>
patients on PD is small and only of borderline signif- 0.05) (Figure 5).
icance (PR = 1 .06; P 0.0 1 ) (Figure 3), whereas in Infection. No significant difference was found be-
patients treated with HD, males have a 25% higher DR tween the DR for all males versus all females due to
than females (RR = 1 .25; P < 0.00 1). Although there is infection (Figure 2). However, further analysis showed
a statistically significant interaction by race (P = that among HD-tneated patients, the DR due to infec-
0.006), the difference in M/F RR is unlikely to be tion was significantly greater for males than for fe-
clinically important (PR = 1 .23 for blacks and RR = males (RR = 1 . 1 1 ; P < 0.00 1 ), whereas among PD-
1 .20 for whites). Interactions of gender by age, cause treated patients, there was no significant difference in
of ESRD, and time on dialysis (“incident” versus “non- infection DR by gender (Figure 6a). There was a 19%
incident”) were not significant for all cause mortality. higher DR for males compared with females among

40 Death Rate (per 100 patient years)

10 Death Rate (per 100 patient years)
RR (M/F) is significantly different The RR (M/F) is significantly different
Female ::.i Male for CAPD V8. HD, pO.001” Female LJ Male in Diabetics vs. Others, p-0.005

30 6.82
24.7 23.7

10 RR1.O6 2.10
p#{149}O.01 2
p’O.OOl’ po.ij
01 0 -
PD HD Diabetes Other Causes
Figure 3. Overall DR due to all causes for PD and HD with Figure 4. Comparison of the M/F DR due to acute Ml for
comparison by gender. DR are for patients with average patients with and without diabetes as a cause of ESRD. DR
characteristics of the study population. RR, DR ratio (M/F), are for patients with average characteristics of the study
adjusted for age, race, diabetes, dialysis modality, and population. RR, DR ratio (M/F), adjusted for age, race,
duration of dialysis (< or > 1 yr). Significant using criteria diabetes, dialysis modality, and duration of dialysis (< or>
outlined in Methods. Significant at the 0.05 level with
** 1 yr). Significant using criteria outlined in Methods. Sig- * *

Bonferroni correction. nificant at the 0.05 level with Bonferroni correction.

1234 Volume 5 . Number 5 . ]994

Bloembergen et al

P = 0.005), and almost three times as likely to with-

10Deat Rate (per 100 patient years) draw from dialysis (BR = 2.72; P < 0.00 1 ) than blacks.
Differences in DR due to cerebrovascular disease and
malignancy were negligible and not statistically signif-
Subgroup analyses were done to determine if these
W/B differences for total deaths on for each cause
varied by other covariates. These results are summa-
nized in Table 3 . Significant interactions are discussed
Deaths due to All Causes. The all-cause W/B RR
. Male --
Female was significantly accentuated (P < 0.00 1 ) in patients
with diabetic ESRD (P < 0.00 1 ) and in “incident”
25 30 35 40 45 50 55 60 65 70 75 80 patients (P < 0.001). The W/B RR for patients with
Age diabetes was 1 .54 (P < 0.00 1 ), whereas it was only
1 . 1 7 (P < 0.00 1 ) in patients with ESRD due to causes
Figure 5. DR due to acute Ml by age for males and females.
DR are for patients with average characteristics of the study other than diabetes (Figure 9a). The W/B RR for
population. RR, DR ratio (M/F), adjusted for age, race, patients treated for less than 1 yr was 1 .4 (P < 0.001)
diabetes, dialysis modality, and duration of dialysis (< or> and was 1 .25 (P < 0.00 1 ) in patients treated for more
1 yr). DR different; P < 0.05. DR different; P < 0.01 (P < than 1 year (Figure 9b). The W/B difference in DR due
0.008 would be significant at the 0.05 level with Bonferroni to all causes increased significantly with age (P =

correction). 0.00 1 ), with the DR for whites becoming significantly

greaten than that for blacks above age 35 (P < 0.01).
Acute MI. As for the all-cause DR. the increased
blacks (RR = 1 . 1 9; P < 0.00 1), although there was no W/B PR due to acute MI was significantly accentuated
gender difference among whites (Figure 6b). (P < 0.00 1 ) in patients with diabetic nephropathy (PR
Malignancy. The M/F RR of 1 .59 due to malignancy = 1 .6 1 ; P < 0.00 1 ) compared with patients with ESRD
was significantly accentuated in older patients (P =
due to other causes (PR = 1 .20; P < 0.001).
“Other Cardiac Causes.” Similarly, the W/B RR was
Cerebrovascular Disease, Withdrawal From Dialy-
significantly accentuated (P < 0.001) in patients with
sis and “Other Cardiac Causes.” The M/F RR for
diabetic nephnopathy (PR = 1.56; P < 0.001) com-
these causes (Figure 2) were not significantly modified
pared with other causes of ESRD (PR = 1 . 18; P <
by age, race, cause of ESRD, dialysis modality, and
0.00 1) and in “incident” patients (RR = 1 .53; P <
time on dialysis (“incident” versus “nonincident”)
0.00 1 ) compared with “nonincident” (PR = 1 .25: P <
when the Bonfennoni correction was applied.
Infection. Although the DR due to infection was
Race higher in whites compared with blacks, this difference
was noted only in patients with diabetic nephnopathy
There were 12,536 deaths occurring among blacks
(PR = 1 .22; P < 0.00 1 ) and not in patients with other
and 28,458 deaths occurring among whites during the
causes of ESRD (PR = 1 .03; P = 0.48). Race differ-
study period. The white-to-black (W/B) RR was 1 .29 (P
ences existed among females (RR = 1 . 1 8; P < 0.001)
< 0.00 1 ), indicating that the DR among whites is 29%
higher than among blacks adjusted for age, gender, but not men (PR = 0.99; P = 0.88). Death rates by
cause of ESRD (diabetic versus others), dialysis mo- gender are shown in Figure 6b.
dality, and length of time on dialysis (“incident” versus Cerebrovascular Disease. Although the W/B PR
“nonincident”). The estimated DR for patients with the due to cenebrovascular disease was overall not signif-
average characteristics of the total population was icantly different from 1 .0, further analysis revealed
24. 1 among whites and was 18.7 among blacks (Fig- that, in patients with diabetic nephnopathy, whites
ure 7). had a significantly greaten risk (PR = 1 .28; P < 0.001),
Figure 8 shows the DR for each individual cause of whereas in patients with other causes of ESRD, there
death and compares the DR for whites and blacks for was no significant difference In the death rate between
each cause of death. As in the male and female blacks and whites (PR = 0.89: P > 0.01) (Figure 10).
subgroups, “other cardiac causes” had by far the Withdrawal From Dialysis and Malignancy. The
highest death rate, followed by acute MI, infection, large W/B DR ratio (RR = 2.72) for withdrawal and the
cerebnovascular disease, withdrawal, and malignancy DR ratio for malignancy (PR = 1 .00), in subanalyses,
in decreasing order. Whites were 34% more likely to did not differ significantly by age, race, cause of ESRD,
die of an acute MI (RR = 1 .34; P < 0.00 1), 30% more or dialysis modality or among “incident” versus “non-
likely to die of “other cardiac causes” (RR = 1 .30; P < incident” patients when the Bonfernoni connection was
0.001), 9% more likely to die ofinfection (RR = 1.09; applied.

Journal of the American Society of Nephrology 1235

Causes of Death in Dialysis Patients

6a. 6b.

PD HD White Black
Figure 6. Comparison of male and female DR due to infection by subgroups: (a) by modality (CAPD and HD) and (b) by race.
DR are for patients with average characteristics of the study population. RR, DR ratio (M/F), adjusted for age, race, diabetes,
dialysis modality, and duration of dialysis (< or > 1 yr). Significant using criteria outlined in Methods. Significant atthe 0.05 * *

level Death Rate (per 100

with Bonferroni patient years)

Death Rate (per 100 patient years)

40 10

Blacks Whites

20 18.7

4 *

p’O.OOl’ 2

Black White Ml Oth.Cardiac CVD Infection Malia Withdrawal

Figure 8. Comparison of DR for each individual cause of

Figure 7. Adjusted all-cause DR for blacks and whites. DR are
death between blacks and whites. DR are for patients with
for dialysis patients with the average characteristics of the
average characteristics of the study population. RR, DR ratio
total study population. RR, DR ratio (W/B), adjusted for age,
(W/B), adjusted for age, race, diabetes, dialysis modality,
gender, diabetes, dialysis modality, and duration of dialysis
(< or > 1 yr).
cardiac causes other than acute Ml; CVD, cerebrovascular
disease; Malig, malignancy; Withdrawal, withdrawal from
dialysis. P < 0.05. < 0.008 (significant
** at the 0.05 level
DISCUSSION with Bonferroni correction).

In all subgroups analyzed in this study (males,

females, whites, blacks), DR due to “other cardiac 515, hyperparathyroidlsm, f3-2-mlcroglobulinemla (18),
causes” were the highest. This category includes con- and complement activation ( 19).
gestive heart failure, subacute and chronic ischemic DR due to infection were similar to those due to
heart disease, cardiomyopathy, arrhythmias, valvular acute MI. Dialysis patients are known to have a high
disease, and penicarditis. DR due to acute MI were incidence of infection due common bacterial patho-
either the second or third highest, depending on the gens as well as opportunistic infections, which are
subgroup. The high prevalence of cardiac disorders in more prevalent in the dialysis patient. Potential con-
ESRD patients has been noted previously (7,14,15) tnibuting factors include altered host defense systems
and has been shown to be associated with an elevated due to uremia and the dialysis procedure ( 19-25), the
risk of mortality ( 1 6, 1 7). Proposed etiologic factors transcutaneous access necessary for both HD and PD
include chronic volume overload, chronic hyperten- (26), and poor nutrition (14).
sion, anemia, arteniovenous fistula, metabolic acido- Withdrawal from dialysis constituted the fourth

1236 Volume 5 ‘ Number 5 ‘ 1994

Bloembergen et al

TABLE 3. Adjusted W/B PR of death for each cause of death by race, cause, modality, and time on dialysis

Gender Cause of ESRD Modality Time on Dialysis

Cause of Death Overall
Male Female Nondiabetes Diabetes HD PD Incident Prevalent

Total 1.29#{176} 1.26#{176} 1.31#{176} 1.17#{176} 1.54#{176} 1.30#{176} 1.20#{176} 1.40 1.25#{176}
Ml 1.34#{176} 1.37#{176} 1.32#{176} 1.20#{176} 1.61#{176} 1.34#{176}1.34#{176} 1.54#{176} 1.29#{176}
Other Cardiac 1.30#{176} 1.35#{176} 1.27#{176} 1.18#{176} 1.56#{176} 1.32#{176} 1.22c 1.53#{176} 1.25#{176}
Cerebrovascular Disease 1.02 1.01 1.03 0.89 1.28#{176} 1.01 1. 17 0.94 1.04
Withdrawal 2.72#{176} 2.71#{176} 2.74#{176} 2.60#{176} 2.96#{176} 2.74#{176} 2.54#{176} 2.71#{176} 2.73#{176}
Malignancy 1.00 1.02 0.97 0.99 1.03 0.99 1.11 1.13 0.95
Infection 1.09c o. 1.18#{176} 1.03 1.22#{176} 0.95 1.10 1.08c

0 Different from 1.00; p < 0.001.

b Boldface areas represent PR that are significantly different from each other (interaction is present) with Bonferroni correction (P < 0.01).
C Different from 1.00; P < 0.01.

9a. 9b.
Death Rate (per 100 patient years) Death Rate (per 100 patient years)
PR (W/B) is significantly different for
EJBIck patients treated or . I year, p’0.001’ [ si.c WhIte j
The RR (W/B) is significantly different
in Diabetics vs. Others, pO.001
- 24.6

Figure 9. Comparison

of white and black DR due to acute Ml by subgroups:
(a) by cause of ESRD and (b) for patients treated
lyear lyear

< or > 1 yr. DR are for patients with average characteristics ofthe study population. PR, DR ratio (W/B), adjusted for age, gender,
cause of ESRD (diabetes versus other causes), modality, and duration of dialysis (< or > 1 yr). Significant according to criteria
outlined in Methods. Significant at the 0.05 level with Bonferroni
* * correction.

highest DR in all subgroups except for blacks. It has amining this by modality reveals that this is only true
previously been shown to be a common cause of death among HD-treated patients. All-cause mortality rates
(9, 10, 13,27-29), accounting for up to 22% ofdeaths in among PD-treated females are statistically only mar-
one study ofpredommnantly white ESRD patients (27). gmnally different than among males and similar to
Studies have suggested that it is increasing as a cause HD-treated males. Nolph et al. have previously docu-
of death in the dialysis population (9,30). mented the absence of a gender effect on mortality in
DR due to malignancy were low relative to other PD patients (35). In PD-treated patients, males had
major causes ofdeath; however, previous studies have higher DR due to MI, other cardiac causes, and ma-
suggested an increased risk of malignancy (3 1-33) on lignancy but lower DR due to cerebnovasculan disease,
ofspecific malignancies (34) in the ESRD as compared withdrawal from dialysis, and infection, with the oven-
with the general population. An increased risk of all result ofa small increased risk ofmontality in males
malignancy in ESRD patients may be due to the effect that is ofbonderline significance. The M/F RR were in
of therapy ( 19,25), the presence of uremia, or the general higher among HD-tneated than among PD-
selection to dialysis of patients with an underlying treated patients for all causes ofdeath; however, these
malignancy (7). were only significantly different for infection (Table 2).
The higher risk of death in male as compared with
Effect of Gender
female dialysis patients appears to be due to an excess
Similar to other studies (4,7), this study has con- of deaths from acute MI, other cardiac causes, and
firmed that the “all-cause” DR is lower for females malignancy. For deaths due to acute MI, gender had a
than for males, when adjusted for race, age, cause of smaller effect among patients with diabetic nephnop-
ESRD, dialysis modality, and incidence. However, ex- athy than among patients with causes of ESRD other

Journal of the American Society of Nephrology 1237

Causes of Death in Dialysis Patients

Death Rate (per 100 patient years) increasing age (where transplantation is less frequent)
suggests that it is not a strong possibility. It should
Black Li White also be noted that, despite this possible selection of
4 The RR (W/B) is significantly different
presumably healthy males to transplantation, males
in Diabetics vs. Others, pO.001’
still have a higher mortality rate than females in the
general ESRD (dialysis and transplant) population (1).
3. 2.77
(3) Treatment. Differences in treatment could also
explain the difference in DR in males versus females.
2 216 Degoulet et a!. ( 1 2) have shown that cardiovascular
1.39 24 deaths are higher in patients dialyzed twice weekly as
1 pR1T;i RR089 j compared with those dialyzed three times weekly. This
p’0.03 J. : 4 suggests that the high premature incidence of cardiac
disease in ESRD patients compared with the general
0 L I ._.._J
population could be the result of an inadequately
Diabetes Other Causes dialyzed unemic toxin that is detrimental to the heart.
If the dialysis prescription is not individualized, fe-
Figure 10. Comparison of differences in the W/B DR due to
males, because of their smaller body size, may be
cerebrovascular disease by subgroups by cause of ESRD. DR
more likely to be adequately dialyzed than males. A
are for patients with average characteristics of the study
population. RR, DR ratio (W/B), adjusted for age, gender, recent USRDS study has actually confirmed such
cause of ESRD (diabetes versus other causes), modality, and gender differences In delivered KT/V among hemodi-
duration of dialysis (< or > 1 yr). Significant according to alysis patients (P.J. Held, et al. , personal communica-
criteria outlined in Methods. Significant at the 0.05 level tion). The smaller gender effect among PD-treated
with Bonferroni correction. patients may suggest that these dialysis dose differ-
ences are smaller in PD- than in HD-treated patients.
(4) Biologic Factors. Biologic differences may ex-
than diabetes, although as expected, the DR due to
plain the differences in mortality by gender. In the
acute MI infarction are much higher overall in diabet-
general population, many diseases affect gender dif-
ics than in nondiabetics (Figure 4). The excess risk of
death for males due to these identified causes requires fenentially, and overall, the risk of death is also higher
the consideration of a number of potential explana- for men than for women. For example, coronary artery
tions, including the following. disease is more common overall in males than in
(1) Selection Resulting From Differential Comor- females. Vital Statistics data indicate that, in 1988,
bidity. Selection could lead to higher mortality among the overall age-adjusted DR due to MI was more than
males, if males with comorbid conditions such as twice as high for males as for females (8 1 .6 versus
coronary artery disease, other cardiac disease, or 38.8/ 100,000 population; PR = 2. 10) (39). This PR
underlying malignancy were more likely to be offered decreases with age, in part because of the loss of the
dialysis therapy than females. Likewise, if females protective effect of estrogen against coronary artery
with any of these conditions were more likely to refuse disease in postmenopausal women. The M/F PR
dialysis therapy, similar differences in mortality noted in the dialysis population of 1 .48 is lower than
would result. To our knowledge, no studies have in the general population. This may reflect the large
specifically examined this issue. The smaller gender proportion of postmenopausal women in the ESRD
effect present among PD-treated patients could result population and/or the high incidence of ovarian dys-
if males selected to PD were relatively healthier than function seen in pnemenopausal females with ESRD.
women selected to this modality. The influence of The M/F PR (adjusted for age) for “major cardiovas-
these factors and their possible interactions on mo- cular disease other than acute MI” was 1 .62 in the
dality choice deserve further study. general population compared with an PR of 1 .30 in
(2) SelectIon Resulting From Differential Trans- this study of dialysis patients. The M/F PR for malig-
plantation Rates. It has been shown that transplan- nant neoplasms in the general population was 1.46
tation rates are higher in males than in females, compared with 1 .59 in this dialysis population.
controlling for age and race ( 1 ,36-38). Because the It therefore appears that similar biologic factors may
presence of cardiac disease is a surgical risk and the be contributing to mortality in the ESRD population
presence of malignancy is a contraindication to trans- as in the general population; however, it must be
plantation, patients with these conditions are less emphasized that DR in the ESRD population are
likely to receive a transplant. Ifmales were more likely much higher than in the general population. This
to receive transplants than females, then the propor- suggests that these disease processes are greafly ac-
tion of males remaining on dialysis with cardiac dis- celenated before and/or after the onset of ESRD.
ease on malignancy would be higher than the pnopor- (5) Social/Cultural. Social or cultural differences
tion of females with these comonbid conditions. existing between males and females may affect mor-
Although this explanation requires consideration, the tality rates. For example, female dialysis patients have
absence of a significant decrease in the M/F PR with been shown to be more compliant than their male

1238 Volume 5 ‘ Number 5 ‘ 1994

Bloembergen et al

counterparts (S.L. Bame, et a!. , personal communica- among “incident” patients compared with “noninci-
tion), which may in turn affect mortality. dent” patients, which is likely the chief contributor to
Among patients treated with PD, females were more a similar observation in the “all-cause” DR compani-
likely to withdraw from dialysis than males. Although son.
this was not significant, nor was it significantly differ- A number of potential explanatory factors for the
ent than the PR among HD-treated patients, it may observed excess risk of death for whites compared
reflect general differences in social/cultural attitudes with blacks require consideration. These include the
among males and females that may influence other following.
treatment decisions and, as a result, affect mortality. (1) Selection Resulting From Differential Comor-
Further studies may be helpful to determine which of bidity. Many of the possible explanations discussed
these explanations related to selection, treatment, previously for males that relate to selection may also
biology, or social/cultural differences are the major apply when considering the excess of mortality seen in
contributors to the excess DR seen in males. whites, including higher dialysis acceptance rates
among whites than blacks with comonbid conditions
and/or a greaten tendency for blacks with serious
Effect of Race
comonbid conditions to refuse therapy. In addition,
The “all-cause” adjusted DR was higher for whites there is concern that blacks have less access to med-
than for blacks in this national study of dialysis ical care in general. For example, recent studies have
patients, confirming a number of previous reports found that whites are more likely than blacks to
(5,6). Crude mortality rates as reported by the USRDS undergo invasive cardiac procedures (4 1 ) and coro-
indicate that DR for black ESRD patients are higher nary artery bypass grafting (42).
than for whites to approximately age 40, appear sim- (2) Selection Resulting From Differential Trans-
ilar to whites from age 40 to 55, and are lower than plantation. Whites have higher transplantation rates
whites above age 55 (2). In this adjusted analysis of than do blacks when controlling for age and gender
dialysis-only patients, DR in whites were significantly (36-38), selectively leaving a greater proportion of
higher than in blacks above the age of 35 yr and whites with comonbid conditions including cardiac
Increased with age but were not significantly different disease on perhaps infection (i.e. , unacceptable sungi-
in patients less than age 35. This apparent discnep- cal risk) on dialysis. However, if this were an explana-
ancy with the crude USRDS data could represent the tion for the increased mortality in whites, one would
effect of adjustment or the exclusion of transplant expect the W/B PR to decrease with age because
patients in our analysis. Although the USRDS Case- transplantation is less frequent in the elderly. In fact,
Mix study did not find a significant difference in it increased with age in this study, suggesting that
mortality by race, there was a trend to higher mortal- this explanation is not a strong possibility.
ity in whites (PR = 1 . 18), consistent with our results. (3) Treatment. Although a difference in treatment
The smaller sample size, difference in patients studied would be a potential explanation, it is less plausible
(all incident rather than incident and prevalent), that the racial difference in DR is due to a treatment
and/on adjustment for comorbid conditions may ac- difference favoring blacks, unless perhaps blacks on
count for the failure to find this risk difference to be dialysis are more compliant than whites. To our
significant in the Case-Mix study. knowledge, there are no data to suggest that blacks
These findings of higher mortality among white receive a higher dose of dialysis.
dialysis patients are opposite to those in the general (4) Biology. Although race has not been found to
population, where blacks have a mortality risk that is designate important genetic factors, physiologic dif-
higher relative to whites (39). Cardiovascular diseases ferences have been described between blacks and
have been shown to be the single largest contributing whites, including differences in nenin levels (43-45),
factor to the mortality gap between blacks and whites sodium-lithium cotransport (46,47), and autonomic
in the general population (40). reactivity (48-50). It is conceivable that differences in
The higher risk of death in whites compared with other physiologic factors may exist that are protective
blacks in this study of ESRD patients is largely due to in the black ESRD population.
an excess of deaths from acute MI and other cardiac (5) Social/Cultural. Many social and cultural differ-
causes, infection (only among females and patients ences exist between blacks and whites. It is conceiv-
with diabetic nephropathy as cause of ESRD), and able that these differences may in some way play a
withdrawal from dialysis. The race effect for deaths protective role in the black ESRD population. For
due to both cardiac categories combined (acute MI example, blacks in the United States have been shown
and other cardiac causes) was noted to be approxi- to consume fewer dairy products than whites (51),
mately three times greater in patients with ESRD due perhaps allowing better dietary compliance and pos-
to diabetic nephropathy as compared with those with sible improved outcome among blacks. In addition, a
ESRD due to other causes. This can account for the higher prevalence of obesity has been found among
greater race effect noted in diabetics in the total blacks (52). Of interest is a USRDS finding of lower
(all-cause) DR comparison. For both cardiac catego- mortality associated with obesity in incident hemodi-
ries, the race effect was also two to three times greater alysis patients (7).

Journal of the American Society of Nephrology 1239

Causes of Death in Dialysis Patients

Social on cultural differences may in part contribute sification of the follow-up interval into shorten versus
to the almost threefold higher risk of death due to longer ESRD duration. To determine if there was a
withdrawal in whites than in blacks, also found pre- difference in risk among “incident” and “nonincident”
viously in a study of dialysis patients in Michigan (9). patients with respect to the associations of interest,
Speculations as to the explanation of this fmding interaction models by duration of dialysis (“incident”
include a higher acceptance of white patients likely to versus “nonincident”) were performed. These mnterac-
withdraw, differences in cultural attitudes on religious tion models were not significant in the analysis of
beliefs toward the discontinuation of therapy, or per- effect of gender on overall on cause-specific mortality,
haps a lesser degree of trust of black patients and suggesting that the M / F differences observed are not
their families toward their predominantly white phy- significantly different for patients with shorten versus
sicians. longer ESRD duration. The effect of race (mortality
higher in whites) on all-cause mortality and on “other
Effect of Gender and Race cardiac causes” was accentuated (significantly
greater) in the “incident” population, suggesting that a
The all-cause DR for men versus women did not
study of incident-only patients would be likely to yield
vary to an important degree by race. Likewise, the
results in the same direction. Nevertheless, a compar-
overall relative DR for whites versus blacks did not
Ison of cause of death by gender and race among
vary by gender. Black females had the lowest all-cause
incident dialysis patients for whom individual data are
DR. whereas white males had the highest. With the
available would allow a more detailed analysis of the
exception of infectious causes, the effect of gender did
role of time on dialysis.
not differ significantly for blacks compared with
This study has also controlled for other known or
whites for individual causes of death, non was the
possible risk factors on gender- and race-specific mor-
effect of race different between males and females. DR
tality including age, cause of ESRD, modality, and
due to infection were similar for all groups, except
race (in the gender comparison) or gender (in the race
black females, for whom they were significantly lower.
comparison). However, there are likely to be other
Again, differences in selection, treatment, biologic
differences between the subgroups compared that
characteristics, and social or cultural factors must be
may be associated with mortality and that have not
considered as possible explanatory factors for these
been measured on adjusted for in this study. For
example, differences in comonbidity, nutritional sta-
tus, and dose of dialysis (previously shown to be
associated with mortality), as well as compliance,
In order to ensure data on a large number of dialysis education, social circumstances, and patients’ health
patient deaths, for this study, a prevalent study pop- attitudes and beliefs, may exist between men and
ulation was evaluated. A prevalent sample includes a women or between blacks and whites. Differential
spectrum of patients in terms of duration of prior compliance and dose of dialysis between men and
therapy as distinct from an incident sample. Studies women have previously been described; however, to
including prevalent patients may be limited by the our knowledge, there are few published data regarding
inherent selection of patients who are survivors. It racial or gender differences in these other factors. In
could be argued that significant relationships noted in addition, differences in facility features and physician
prevalent patients only may be suspect for such “sur- training or practices in predominately white versus
vivon selection bias.” To address this issue, we used an predominately black dialysis units could bias racial
indicator variable, time on dialysis (“incident” on “non- comparisons. The role of these various factors de-
incident”), as discussed in Methods. Ideally, follow-up serves further study.
time would be classified for each patient on the basis Patient follow-up is likely to be very incomplete
of the exact dates of ESRD therapy start for that during the first 90 days because Medicare coverage
patient, and a consistent definition of the period of does not start until after this time period. As outlined
time to be labeled as “incident” could be adopted. in Methods, those patients starting dialysis less than
However, because of the tabular data available to us, 3 months before January 1 were excluded for that year
we could not classify the follow-up time for each to avoid the bias that might be introduced by basing
patient during a year according to the exact duration an analysis on the distinguished patients whose data
of ESRD for that patient, but had to rely on a classi- are available during the first 90 days. In addition, the
fication of the duration at the start of the year. Our cohort scheme used did not allow the capture of data
classification system has some imprecision because on incident patients who did not survive to the first of
follow-up time intervals labeled as “incident” can con- a calendar year. If the effect of gender on race was
respond to an ESRD duration ranging between 3 and substantially different in these excluded patients than
24 months, whereas follow-up time intervals labeled in those included in the study, the overall effect of
as “nonincident” can correspond to ESRD duration gender on race may be quite different. In fact, there
ranging as low as 12 months. Despite the overlap in was no significant difference in the M/F PR of all-
underlying ESRD duration caused by the tabular data cause mortality between the “incident” and “noninci-
available to us, our definitions do yield a rough clas- dent” patients that were included in our study; more-

1240 Volume 5 . Number 5 ‘ 1994

Bloembergen et al

over, in the comparison ofmortality by race, there was tality rates among dialysis treatment centers. Ann Intern
Med l992;1 17:332-336.
accentuation of the observed W/B PR among the
7. U.S. Renal Data System. USRDS 1992 Annual Data
“incident” patients included. These findings suggest Report. Comorbid conditions and correlations with mar-
that the exclusion criteria of some incident patients tality risk among 3,399 incident hemodialysis patients.
Am J Kidney Dis l992;2OlSuppl 2J:32-38.
are unlikely to substantially alter the overall effect of
8. Mailloux LU, Belluci AG, Wilkes BM, et at.: Mortality in
gender and race on mortality. However, the results of dialysis patients: analysis of the causes of death. Am J
this study should rightfully be generalized only to Kidney Dis 199 l;l8:326-335.
patients similar to those included. 9. Port FK, Wolfe RA, Hawthorne VM, Ferguson CW: Dis-
continuation of dialysis therapy as a cause of death. Am
The performance of this analysis with a large na- J Nephrol l989;8:l45-149.
tional data set has obvious advantages, including the 10. Port FK: Mortality and causes of death in patients with
enhancement of statistical power as well as the gen- end-stage renal failure. Am J Kidney Dis 1990; 15:2 15-
eralizability to the United States. A limitation of a 1 1. Held PJ, Levin NW, Port FK: Cardiac disease in chronic
study of this type is that the cause of death is reported uremia: An overview. In: Parfrey PS, Harnett JD, Eds.
by renal physicians in a manner that has not been Cardiac Dysfunction in Chronic Uremia. Dordrecht: Klu-
wer Academic Publishers; 1992:3-17.
standardized. Despite the listing of23 causes of death
12. Degoulet P, Legrain M, Reach I, et a!.: Mortality risk
on the ESRD Death Notification Form, different phy- factors in patients treated by chronic hemodialysis: Re-
sicians may use different definitions/criteria for each port of the Diaphane Collaborative Study. Nephron
cause. This raises the issue ofmisclassification. Study l982;3 1:103-110.
13. Roberts JC, Kjellstrand CM: Choosing death: With-
results would be compromised if there was a selective drawal from chronic dialysis without medical reason.
difference in misclassification between the groups Acta Med Scand 1988;223:18l-186.
compared. One may argue that this is unlikely, how- 14. Churchifi DN, Taylor DW, Cook RJ, et at.: Canadian
Hemodialysis Morbidity Study. Am J Kidney Dis 1992;
even for example, there is a growing literature suggest- 19:214-234.
ing less aggressive treatment of coronary artery dis- 15. Parfrey PS, Harnett JD, Barre PE: The natural history of
ease in blacks and females. It follows that coronary myocardial disease in dialysis patients. J Am Soc Neph-
rol 1991;2:2-l2.
artery disease may also be less frequently diagnosed 16. Parfrey PS, Harnett JD, Gniffiths SM, et al.: The clinical
in these subgroups, and thus, their deaths would be course of left ventricular hypertrophy in dialysis pa-
less likely attributed to this cause. tients. Nephron 1990;55: 114-120.
1 7. Parfrey PS, Griffiths SM, Harnett JD, et at.: Outcome of
As with most studies evaluating cause of death
congestive heart failure. Heart disease in dialysis pa-
statistics, these limitations restrict this study to being tients. Am J Nephnol 1990; 10:213-221.
one that may be used to generate hypotheses that can 18. Parfrey PS: Cardiac and cerebrovascular disease in
be evaluated by future detailed epidemiologic and chronic uremia. Am J Kidney Dis l993;2 1:77-80.
19. RM: Clinical sequelae of complement activation
clinical studies. The need for prospective studies that in hemodialysis. Clin Nephnol 1986;26:S9-S12.
use standardized definitions and criteria for each 20. Khan IH, Catto GRD: Long-term complications of dialy-
cause of death and that are rigorously designed to 515: Infection. Kidney Int 1993;43:S143-S 148.
2 1 . Descamps-Latscha B, Herbelin A: Long-term dialysis
eliminate threats of biases is emphasized.
and cellular immunity: A critical survey. Kidney Int
ACKNOWLEDGMENTS 22. Vanholder R, Ringoir S: Polymorphonuclean cell func-
tion and infection in dialysis. Kidney Int l992;42:S9l-
WE. Bloembergen is supported by a fellowship award from the
Kidney Foundation of Canada. The data reported herein have been 23. Vanholder R, Ringoir 5: Infectious mortality and defects
supplied by the United States Renal Data System (USRDS). The of phagocytic function in end-stage renal disease: A
interpretation and reporting of these data are the responsibility of the review. J Am Soc Nephrol 1993;3: 1541-1554.
authors and in no way should be seen as an official policy or 24. Zaoui P, Green W, Hakim RM: Hemodialysis with cu-
Interpretation of the USRDS. The authors gratefully acknowledge prophane membrane modulates interleukin-2 receptor
helpful reviews of the manuscript by A. Ojo, MS. . M.D. , and A.A.S. expression. Kidney Int 1991 39:1020-1026.
Lopes. M.D. 25. Zaoui P, Hakim RM: Natural killer-cell function in he-
modialysis patients: Effect of the dialysis membrane.
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Eigenvectors are a special set of vectors associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic vectors, proper vectors, or latent vectors (Marcus and Minc 1988, p. 144).

The determination of the eigenvectors and eigenvalues of a system is extremely important in physics and engineering, where it is equivalent tomatrix diagonalization and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvector is paired with a corresponding so-called eigenvalue. Mathematically, two different kinds of eigenvectors need to be distinguished: left eigenvectors and right eigenvectors. However, for many problems in physics and engineering, it is sufficient to consider only right eigenvectors. The term "eigenvector" used without qualification in such applications can therefore be understood to refer to a right eigenvector.

The decomposition of a square matrix

into eigenvalues and eigenvectors is known in this work as eigen

decomposition, and the fact that this decomposition is always possible as long as the matrix consisting of the eigenvectors of is square is known as the eigen decomposition theorem.

Define a right eigenvector as a column vector

is a matrix, so

which means the right eigenvalues must have zero determinant, i.e.,

Similarly, define a left eigenvector as a row vector

Taking the transpose of each side gives

which can be rewritten as

Rearrange again to obtain

which means

Rewriting gives

(9) (10) (11)

where the last step follows from the identity

(12) Equating equations ( ) and (11), which are both equal to 0 for arbitrary that and , therefore requires

, i.e., left and righteigenvalues are equivalent, a statement that is not true for eigenvectors.

be a matrix formed by the columns of the right eigenvectors and

be a matrix formed by the rows of the

left eigenvectors. Let

(14) (15)

(16) (17)

But this equation is of the form

is a diagonal matrix, so it must be true that

is also diagonal. In particular, if

is a symmetric

matrix, then the left and right eigenvectors are simply each other's transpose, and if is Hermitian), then the left and right eigenvectors areadjoint matrices.

is a self-adjoint matrix (i.e., it

Eigenvectors may not be equal to the zero vector. A nonzero scalar multiple of an eigenvector is equivalent to the original eigenvector. Hence, without loss of generality, eigenvectors are often normalized to unit length.

While an between 0 and eigenvector

matrix always has

eigenvalues, some or all of which may be degenerate, such a matrix may have has only the single

linearly independent eigenvectors. For example, the matrix .

Eigenvectors may be computed in Mathematica using Eigenvectors[matrix]. This command always returns a list of length , so any eigenvectors that are not linearly independent are returned as zero vectors. Eigenvectors and

eigenvalues can be returned together using the command Eigensystem[matrix].

Given a

with eigenvectors

, and

and corresponding eigenvalues

, and

, then an

arbitrary vector

can be written

Applying the matrix

, and

, it therefore follows that

so repeated application of the matrix to an arbitrary vector amazingly results in a vector proportional to the eigenvector with largest eigenvalue.

Let A be a square matrix of order n and one of its eigenvalues. Let X be an eigenvector of A associated to . We must have

This is a linear system for which the matrix coefficient is . Since the zerovector is a solution, the system is consistent. In fact, we will in a different page that the structure of the solution set of this system is very rich. In this page, we will basically discuss how to find the solutions. Remark. It is quite easy to notice that if X is a vector which satisfies , then the vector Y = c X (for any arbitrary number c) satisfies the same equation, i.e. . In other words, if we know that X is an eigenvector, then cX is also an eigenvector associated to the same eigenvalue. Let us start with an example. Example. Consider the matrix

First we look for the eigenvalues of A. These are given by the characteristic equation , i.e.

If we develop this determinant using the third column, we obtain

Using easy algebraic manipulations, we get

which implies that the eigenvalues of A are 0, -4, and 3. Next we look for the eigenvectors. 1. Case : The associated eigenvectors are given by the linear system

which may be rewritten by

Many ways may be used to solve this system. The third equation is identical to the first. Since, from the second equations, we have y = 6x, the first equation reduces to 13x + z = 0. So this system is equivalent to

So the unknown vector X is given by

Therefore, any eigenvector X of A associated to the eigenvalue 0 is given by

where c is an arbitrary number. 2.

: The associated eigenvectors are given by the linear system

which may be rewritten by

In this case, we will use elementary operations to solve it. First we consider the augmented matrix , i.e.

Then we use elementary row operations to reduce it to a upper-triangular form. First we interchange the first row with the first one to get

Next, we use the first row to eliminate the 5 and 6 on the first column. We obtain

If we cancel the 8 and 9 from the second and third row, we obtain

Finally, we subtract the second row from the third to get

Next, we set z = c. From the second row, we get y = 2z = 2c. The first row will imply x = -2y+3z = -c. Hence

Therefore, any eigenvector X of A associated to the eigenvalue -4 is given by

where c is an arbitrary number. 2. Case : The details for this case will be left to the reader. Using similar ideas as the one described above, one may easily show that any eigenvector X of A associated to the eigenvalue 3 is given by

where c is an arbitrary number. Remark. In general, the eigenvalues of a matrix are not all distinct from each other (see the page on the eigenvalues for more details). In the next two examples, we discuss this problem. Example. Consider the matrix

The characteristic equation of A is given by

Hence the eigenvalues of A are -1 and 8. For the eigenvalue 8, it is easy to show that any eigenvector X is given by

where c is an arbitrary number. Let us focus on the eigenvalue -1. The associated eigenvectors are given by the linear system

which may be rewritten by

Clearly, the third equation is identical to the first one which is also a multiple of the second equation. In other words, this system is equivalent to the system reduced to one equation 2x+y + 2z= 0. To solve it, we need to fix two of the unknowns and deduce the third one. For example, if we set and , we obtain eigenvector X of A associated to the eigenvalue -1 is given by . Therefore, any

In other words, any eigenvector X of A associated to the eigenvalue -1 is a linear combination of the two eigenvectors

Example. Consider the matrix

The characteristic equation is given by

Hence the matrix A has one eigenvalue, i.e. -3. Let us find the associated eigenvectors. These are given by the linear system

which may be rewritten by

This system is equivalent to the one equation-system x - y = 0.

So if we set x = c, then any eigenvector X of A associated to the eigenvalue -3 is given by

Let us summarize what we did in the above examples. Summary: Let A be a square matrix. Assume is an eigenvalue of A. In order to find the associated eigenvectors, we do the following steps: 1. Write down the associated linear system

2. Solve the system. 3. Rewrite the unknown vector X as a linear combination of known vectors. The above examples assume that the eigenvalue is real number. So one may wonder whether any eigenvalue is always real. In general, this is not the case except for symmetric matrices. The proof of this is very complicated. For square matrices of order 2, the proof is quite easy. Let us give it here for the sake of being little complete. Consider the symmetric square matrix

Its characteristic equation is given by

This is a quadratic equation. The nature of its roots (which are the eigenvalues of A) depends on the sign of the discriminant

Using algebraic manipulations, we get

Therefore, numbers.

is a positive number which implies that the eigenvalues of A are real

Remark. Note that the matrix A will have one eigenvalue, i.e. one double root, if and only if . But this is possible only if a=c and b=0. In other words, we have A = a I 2.

In the next page, we will discuss the case of complex eigenvalues.

Matrix Tutorial 3: Eigenvalues and Eigenvectors

A tutorial on eigenvalues, eigenvectors and their properties. Includes step by step how-to calculations. An introduction to vector iteration, the Power Method and the Deflation Method is provided.
Dr. E. Garcia Mi Email | Last Update: 07/17/06 Topics
Putting Everything Together The Eigenvalue Problem Calculating Eigenvalues Eigenvectors Properties of Eigenvalues and Eigenvectors Computing Eigenvectors from Eigenvalues Computing Eigenvalues from Eigenvectors The Power Method (Vector Iteration) The Deflation Method Why should we care about all this? Beware of Link Model Speculators Tutorial Review References

Putting Everything Together

In Part 1 of this three-part tutorial we defined different type of matrices. We covered digraphs, stochastic matrices, and markov chains. We also mentioned how some search engine marketers have derived blogonomies out of these and similar concepts. In Part 2 we covered matrix operations like addition, substraction and multiplication of matrices. We also discussed multiplication and division of matrices by a scalar and calculation of determinants from square matrices. We mentioned that if a determinant has a nonzero value, its matrix is described as regular and that if a determinant has zero value, its matrix is described as singular. It is now time to put everything together, to demystify eigenvalues, eigenvectors, and present some practical applications.

The Eigenvalue Problem

Consider a scalar matrix Z, obtained by multiplying an identity matrix by a scalar; i.e., Z = c*I. Deducting this from a regular matrix A gives a new matrix A - c*I.

Equation 1: A - Z = A - c*I. If its determinant is zero, Equation 2: |A - c*I| = 0 and A has been transformed into a singular matrix. The problem of transforming a regular matrix into a singular matrix is referred to as the eigenvalue problem. However, deducting c*I from A is equivalent to substracting a scalar c from the main diagonal of A. For the determinant of the new matrix to vanish the trace of A must be equal to the sum of specific values of c. For which values of c?

Calculating Eigenvalues
Figure 1 shows that the computation of eigenvalues is a straightforward process.

Figure 1. The eigenvalue problem.

In the figure we started with a matrix A of order n = 2 and deducted from this the Z = c*I matrix. Applying the method of determinants for m = n = 2 matrices discussed in Part 2 gives

|A - c*I| = c2 - 17*c + 42 = 0 Solving the quadratic equation, c1 = 3 and c2 = 14. Note that c1 + c2 = 17, confirming that these characteristic values must add up to the trace of the original matrix A (13 + 4 = 17). The polynomial expression we just obtained is called the characteristic equation and the c values are termed the latent roots or eigenvalues of matrix A. Thus, deducting either c1 = 3 or c2 = 14 from the principal of A results in a matrix whose determinant vanishes (|A - c*I| = 0) In terms of the trace of A we can write: c1/trace = 3/17 = 0.176 or 17.6% c2/trace = 14/17 = 0.824 or 82.4% Thus, c2 = 14 is the largest eigenvalue, accounting for more than 82% of the trace. The largest eigenvalue of a matrix is also called the principal eigenvalue. There are many scenarios like in Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) in which some eigenvalues are so small that are ignored. Then the remaining eigenvalues are added together to compute an estimated fraction. This estimate is then used as a correlation criterion for the so-called Rank Two approximation.
SVD and PCA are techniques used in cluster analysis. In information retrieval, SVD is used in Latent Semantic Indexing (LSI) while PCA is used in Information Space (IS). These will be discussed in upcoming tutorials.

Now that the eigenvalues are known, these are used to compute the latent vectors of matrix A. These are the so-called eigenvectors.

Equation 1 can be rewritten for any eigenvalue i as Equation 3: A - ci*I Multiplying by a column vector Xi of same number of rows as A and setting the results to zero leads to Equation 4: (A - ci*I)*Xi = 0 Thus, for every eigenvalue ci this equation constitutes a system of n simultaneous homogeneous equations, and every system of equations has an infinite number of solutions. Corresponding to

every eigenvalue ci is a set of eigenvectors Xi, the number of eigenvectors in the set being infinite. Furthermore, eigenvectors that correspond to different eigenvalues are linearly independent from one another.

Properties of Eigenvalues and Eigenvectors

At this point it might be a good idea to highlight several properties of eigenvalues and eigenvectors. The following pertaint to the matrices we are dicussing here, only.
y y y y y y y y y the absolute value of a determinant (|detA|) is the product of the absolute values of the eigenvalues of matrix A c = 0 is an eigenvalue of A if A is a singular (noninvertible) matrix If A is a nxn triangular matrix (upper triangular, lower triangular) or diagonal matrix , the eigenvalues of A are the diagonal entries of A. A and its transpose matrix have same eigenvalues. Eigenvalues of a symmetric matrix are all real. Eigenvectors of a symmetric matrix are orthogonal, but only for distinct eigenvalues. The dominant or principal eigenvector of a matrix is an eigenvector corresponding to the eigenvalue of largest magnitude (for real numbers, largest absolute value) of that matrix. For a transition matrix, the dominant eigenvalue is always 1. The smallest eigenvalue of matrix A is the same as the inverse (reciprocal) of the largest eigenvalue of A-1; i.e. of the inverse of A.

If we know an eigenvalue its eigenvector can be computed. The reverse process is also possible; i.e., given an eigenvector, its corresponding eigenvalue can be calculated. Let's illustrate these two cases.

Computing Eigenvectors from Eigenvalues

Let's use the example of Figure 1 to compute an eigenvector for c1 = 3. From Equation 2 we write

Figure 2. Eigenvectors for eigenvalue c1 = 3. Note that c1 = 3 gives a set with infinite number of eigenvectors. For the other eigenvalue, c2 = 14, we obtain

Figure 3. Eigenvectors for eigenvalue c2 = 14. In addition, it is confirmed that |c1|*|c2| = |3|*|14| = |42| = |detA|. As show in Figure 4, plotting these vectors confirms that eigenvectors that correspond to different eigenvalues are linearly independent of one another. Note that each eigenvalue produces an infinite set of eigenvectors, all being multiples of a normalized vector. So, instead of plotting candidate eigenvectors for a given eigenvalue one could simply represent an entire set by its normalized eigenvector. This is done by rescaling coordinates; in this case, by taking coordinate ratios. In our example, the coordinates of these normalized eigenvectors are:
1. (0.5, -1) for c1 = 3. 2. (1, 0.2) for c2 = 14.

Figure 4. Eigenvectors for different eigenvalues are linearly independent. Mathematicians love to normalize eigenvectors in terms of their Euclidean Distance (L), so all vectors are unit length. To illustrate, in the preceeding example the coordinates of the two eigenvectors are (0.5, -1) and (1, 0.2). Their lengths are for c1 = 3: L = [0.52 + -12]1/2 = 1.12 for c2 = 14: L = [12 + 0.22]1/2 = 1.02 Their new coordinates (ignoring rounding errors) are for c1 = 3: (0.5/1.12, -1/1.12) = (0.4, -0.9) for c2 = 14: (1/1.02, 0.20/1.02) = (1, 0.2) You can do the same and normalize eigenvectors to your heart needs, but it is time consuming (and boring). Fortunately, if you use software packages these will return unit eigenvectors for you by default. How about obtaining eigenvalues from eigenvectors?

Computing Eigenvalues from Eigenvectors

This is a lot easier to do. First we rearrange Equation 4. Since I = 1 we can write the general expression Equation 5: A*X = c*X Now to illustrate calculations let's use the example given by Professor C.J. (Keith) van Rijsbergen in chapter 4, page 58 of his great book The Geometry of Information Retrieval (3), which we have reviewedalready.

Figure 5. Eigenvalue obtained from an eigenvector. This result can be confirmed by simply computing the determinant of A and calculating the latent roots. This should give two latent roots or eigenvalues, c = 41/2 = +/- 2. That is, one eigenvalue must be c1 = +2 and the other must be c2 = -2. This also confirms that c1 + c2 = trace of A which in this case is zero. An Alternate Method: Rayleigh Quotients An alternate method for computing eigenvalues from eigenvectors consists in calculating the socalled Rayleigh Quotient, where Rayleigh Quotient = (XT*A*X)/(XT*X) where XT is the transpose of X. For the example given in Figure 5, XT*A*X = 36 and XT*X = 18; hence, 36/18 = 2. Rayleigh Quotients give you eigenvalues in a straightforward manner. You might want to use this method instead of inspection or as double-checking method. You can also use this in combination with other iterative methods like the Power Method.

The Power Method (Vector Iteration)

Eigenvalues can be ordered in terms of their absolute values to find the dominant or largest eigenvalue of a matrix. Thus, if two distinct hypothetical matrices have the following set of eigenvalues
y y 5, 8, -7; then |8| > |-7| > |5| and 8 is the dominant eigenvalue. 0.2, -1, 1; then |1| = |-1| > |0.2| and since |1| = |-1| there is no dominant eigenvalue.

One of the simplest methods for finding the largest eigenvalue and eigenvector of a matrix is the Power Method, also called the Vector Iteration Method. The method fails if there is no dominant eigenvalue. In its basic form the Power Method is applied as follows:
1. 2. 3. 4. 5. 6. Asign to the candidate matrix an arbitrary eigenvector with at least one element being nonzero. Compute a new eigenvector. Normalize the eigenvector, where the normalization scalar is taken for an initial eigenvalue. Multiply the original matrix by the normalized eigenvector to calculate a new eigenvector. Normalize this eigenvector, where the normalization scalar is taken for a new eigenvalue. Repeat the entire process until the absolute relative error between successive eigenvalues satisfies an arbitrary tolerance (threshold) value.

It cannot get any easier than this. Let's take a look at a simple example.

Figure 6. Power Method for finding an eigenvector with the largest eigenvalue. What we have done here is apply repeatedly a matrix to an arbitrarily chosen eigenvector. The result converges nicely to the largest eigenvalue of the matrix; i.e. Equation 6: AkXi = cik*Xi Figure 7 provides a visual representation of the iteration process obtained through the Power Method for the matrix given in Figure 3. As expected, for its largest eigenvalue the iterated vector converges to an eigenvector of relative coordinates (1, 0.20).

Figure 7. Visual representation of vector iteration. It can be demonstrated that guessing an initial eigenvector in which its first element is 1 and all others are zero produces in the next iteration step an eigenvector with elements being the first column of the matrix. Thus, one could simply choose the first column of a matrix as an initial seed. Whether you want to try a matrix column as an initial seed, keep in mind that the rate of convergence of the power method actually depends on the nature of the eigenvalues. For closely spaced eigenvalues, the rate of convergence can be slow. Several methods for improving the rate of convergence have been proposed (Shifted Iteration, Shifted Inverse Iteration or transformation methods). I will not discuss these at this time. How about calculating the second largest eigenvalue of a matrix?

The Deflation Method

There are different methods for finding subsequent eigenvalues of a matrix. I will discuss only one of these: The Deflation Method. Deflation is a straightforward approach. Essentially, this is what we do:
1. First, we use the Power Method to find the largest eigenvalue and eigenvector of matrix A. 2. multiply the largest eigenvector by its transpose and then by the largest eigenvalue. This produces the matrix Z* = c *X*(X)T 3. compute a new matrix A* = A - Z* = A - c *X*(X)T 4. Apply the Power Method to A* to compute its largest eigenvalue. This in turns should be the second largest eigenvalue of the initial matrix A.

Figure 8 shows deflection in action for the example given in Figure 1 and 2. After few iterations the method converges smoothly to the second largest eigenvalue of the matrix. Neat!

Figure 8. Finding the second largest eigenvalue with the Deflation Method. Note. We want to thanks Mr. William Cotton for pointing us of an error in the original version of this figure, which was then compounded in the calculations. These have been corrected since then. After corrections, still deflation was able to reach the right second eigenvalue of c = 3. Results can be double checked using Raleigh's Quotients. We can use deflation to find subsequent eigenvector-eigenvalue pairs, but there is a point wherein rounding error reduces the accuracy below acceptable limits. For this reason other methods, like Jacobi's Method, are preferred when one needs to compute many or all eigenvalues of a matrix.

Why should we care about all this?

Armed with this knowledge, you should be able to understand better articles that discuss link models like PageRank, their advantages and limitations, when these succeed or fail and why. The

assumption from these models is that surfing the web by jumping from links to links is like a random walk describing a markov chain process over a set of linked web pages. The matrix is considered the transition probability matrix of the Markov chain and having elements strictly between zero and one. For such matrices the Perron-Frobenius Theorem tells us that the largest eigenvalue of the matrix is equal to one (c = 1) and that the corresponding eigenvector, which satisfies the equation Equation 7: A*X = X does exists and is the principal eigenvector (state vector) of the Markov Chain, with elements of X being the pageranks. Thus, according to theory, iteration should enable one to compute the largest eigenvalue and this principal eigenvector, whose elements are the pagerank of the individual pages.

Beware of Link Model Speculators

If you are interested in reading how PageRank is computed, stay away from speculators, especially from search engine marketers. It is hard to find accurate explanations in SEO or SEM forums or from those that sell link-based services. I rather suggest you to read university research articles from those that have conducted serious research work on link graphs and PageRankbased models. Great explanations are all over the place. However, some of these are derivative work and might not reflect how Google actually implements PageRank these days (only those at Google know or should know this or if PageRank has been phased out for something better). With all, these research papers are based on experimentation and their results are verifiable. There is a scientific paper I would like readers to at least consider: Link Analysis, Eigenvectors and Stability, from Ng, Zheng and Jordan from the University of California, Berkeley (5). In this paper the authors use many of the topics herein described to explain the HITS and PageRank models. Regarding the later they write:

Figure 9. PageRank explanation, according to Ng, Zheng and Jordan from University of California, Berkeley Note that the last equation in Figure 9 is of the form A*X = X as in Equation 7; that is, p is the principal eigenvector (p = X) and can be obtained through iterations. After completing this 3-part tutorial you should be able to grasp the gist of this paper. The group even made an interesting connection between HITS and LSI (latent semantic indexing). If you are a student and are looking for a good term paper on Perron-Frobenius Theory and PageRank computations, I recommend you the term paper by Jacob Miles Prystowsky and Levi Gill Calculating Web Page Authority Using the PageRank Algorithm (6). This paper discusses PageRank and some how-to calculations involving the Power Method we have described. How many iterations are required to compute PageRank values? Only Google knows. According to this Perron-Frobenius review from Professor Stephen Boyd from Stanford (7), the original paper on Google claims that for 24 million pages 50 iterations were required. A lot of things have changed since then, including methods for improving PageRank and new flaws discovered in this and similar link models. These flaws have been the result of the commercial nature of the Web. Not surprisingly, models that work well under controlled conditions and free from noise often fail miserably when transferred to a noisy environment. These topics will be discussed in details in upcoming articles. Meanwhile, if you are still thinking that the entire numerical apparatus validates the notion that on the Web links can be equated to votes of citation importance or that the treatment validates the

link citation-literature citation analogy a la Eugene Garfield's Impact Factors, think again. This has been one of the biggest fallacies around, promoted by many link spammers, few IRs and several search engine marketers with vested interests. Literature citation and Impact Factors are driven by editorial policies and peer reviews. On the Web anyone can add/remove/exchange links at any time for any reason whatever. Anyone can buy/sell/trade links for any sort of vested interest or overwrite links at will. In such noisy environment, far from the controlled conditions observed in a computer lab, peer review and citation policies are almost absent or at best contaminated by commercialization. Evidently under such circumstances the link citation-literature citation analogy or the notion that a link is a vote of citation importance for the content of a document cannot be sustained.

Prev: Matrix Tutorial 2: Matrix Operations

Tutorial Review
1. Prove that a scalar matrix Z can be obtained by multiplying an identity matrix I by a scalar c; i.e., Z = c*I. 2. Prove that deducting c*I from regular matrix A is equivalent to substracting a scalar c from the diagonal of A. 3. Given the following matrix,

Prove that these are indeed the three eigenvalues of the matrix. Calculate the corresponding eigenvectors. 4. Use the Power Method to calculate the largest eigenvalue of the matrix given in Exercise 3. 5. Use the Deflation Method to calculate the second largest eigenvalue of the matrix given in Exercise 3.


tag:blogger.com,1999:blog-22093933.post8024400133345488362..comments2019-07-13T17:59:27.854-05:00Comments on Deric's MindBlog: What distinguishes our minds from those of other creatures?Deric Bowndshttp://www.blogger.com/profile/16617204535017208765noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-22093933.post-47242321736086261362009-08-26T16:06:14.807-05:002009-08-26T16:06:14.807-05:00The main feature which opens us new worlds and dis...The main feature which opens us new worlds and distinguishes our minds from our nearest relatives is language, and it requires a larger cerebral cortex. If "to map" and "to reduce" are the basic operations of distributed neural networks, the an increased network leads to greater associations and abstractions. This is what Hauser mentions: deeper associations (combination of ideas) and abstractions (abstract thoughts and symbols).jofrhttp://blog.cas-group.net/noreply@blogger.com


Checking Your Machine with Pronterface

This program is useful for testing the endstops on the Ultimaker 2 series printers. For the Ultimaker 3 series printers, endstop testing is built into the firmware. To test your limit switches on the Ultimaker 3 series printers go to: System   Maintenance   Diagnostics   Limit Switches

What is Pronterface and how do I get it?

Pronterface is a free, downloadable program that allows you to check the functionality of different parts of the printer. To visit Pronterface's download page, click here or go to, click download on the top menu, then click Windows and OSX Binaries. Select the latest version for your operating system.

How do I connect my printer?

Turn your printer on and connect to your computer using the USB cable. In Pronterface, select the correct connection for Port (green in photo below) and change the number following the @ to 250000 (blue in photo below). Then hit connect.

How do I check for XY errors or endstop functionality?

Make sure your print head is in the middle of the printer away from the endstops, and that you have the bed raised up so that the Z stop is not triggered. You then type in the command line M119 and hit send (blue in the photo below).

It should tell you that the X, Y, and Z endstops are open. Then, move the printhead to the back left corner by hand, triggering both endstops, and push the bed to the bottom of the printer, and type of the command line again and see if it says that your endstops are triggered.

If the endstops are reporting accurately, you can move on to the next step. If they're not reporting accurately, then one of your endstops is bad.

The lefthand side of Pronterface has a square/circle section with arrows indicating X and Y movement. Try to move the printhead using the arrows in either the X direction or the Y direction. If it successful in one direction, try the other direction.

If the printer fails in one direction or the other and reads off the XY error, you'll want to remove the electronics cover below your printer and switch the X motor cable and Y motor cable and then repeat the process again to determine whether the failure was caused by the board, or the motor.

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