  Researchers often misinterpret and misrepresent statistical outputs. This
abuse has led to a large literature on modification or replacement of testing
thresholds and $P$-values with confidence intervals, Bayes factors, and other
devices. Because the core problems appear cognitive rather than statistical, we
review simple aids to statistical interpretations. These aids emphasize logical
and information concepts over probability, and thus may be more robust to
common misinterpretations than are traditional descriptions. We use the Shannon
transform of the $P$-value $p$, also known as the binary surprisal or $S$-value
$s=-\log_{2}(p)$, to measure the information supplied by the testing procedure,
and to help calibrate intuitions against simple physical experiments like coin
tossing. We also use tables or graphs of test statistics for alternative
hypotheses, and interval estimates for different percentile levels, to thwart
fallacies arising from arbitrary dichotomies. Finally, we reinterpret
$P$-values and interval estimates in unconditional terms, which describe
compatibility of data with the entire set of analysis assumptions. We
illustrate these methods with a reanalysis of data from an existing
record-based cohort study. In line with other recent recommendations, we advise
that teaching materials and research reports discuss $P$-values as measures of
compatibility rather than significance, compute $P$-values for alternative
hypotheses whenever they are computed for null hypotheses, and interpret
interval estimates as showing values of high compatibility with data, rather
than regions of confidence. Our recommendations emphasize cognitive devices for
displaying the compatibility of the observed data with various hypotheses of
interest, rather than focusing on single hypothesis tests or interval
estimates. We believe these simple reforms are well worth the minor effort they
require.
