# Difference between Wald statistics and effect size in regression models

In some of my regression models (using the rms package), I notice a distinct difference between the result of the Wald statistics (estimated using anova.rms) and the estimated effect size (estimated using summary.rms) and try to consolidate this:

In most instances, there is an obvious relationship (let's assume for simplicity p<0.05 as a significance threshold - it is a bit more complicated of course): significant Wald test, significant effect and vice versa. However, in some instances, I notice that while the Wald statistics suggests no significant association between independent variable and effect, the estimated effect actually is significant (both by estimated p-value and 95% CI).

How does one interpret this? One statistic tells me that it is unlikely that there is an association, the other tells me it does.

Example:

a <- ols(ep ~ sex + bmi + age + exposure, data=df)
anova(a)


...

Wald Statistics          Response: ep

Factor          Chi-Square d.f. P
exposure        6.43     2    0.0402


However:

             Effects              Response : ep

Factor         Low      High    Effect    S.E.     Lower 0.95 Upper 0.95
exposure       -0.56     3.5    0.08      0.03     0.02       0.14

• Can you show some example R output to better help us understand what is going on? – Isabella Ghement Mar 20 at 14:47
• I think you might be better off asking on R-help if you cannot find in the documentation what summary() does since although we all know what Wald statistics are we do not know what the rms package does. – mdewey Mar 20 at 17:00
• If the documentation mentions likelihood ratio or profile likelihood that might give us a clue. – mdewey Mar 20 at 17:01
• summary.rms estimates the inter-quartile effect size, adjusted for all other co-variable. – Gux Mar 20 at 19:52