12  Statistical Inference Review

12.1 Types of Analyses

  • Except for one-sample tests, all tests can be thought of as testing for an association between at least one variable with at least one other variable
  • Testing for group differences is the same as testing for association between group and response
  • Testing for association between two continuous variables can be done using correlation (especially for unadjusted analysis) or regression methods; in simple cases the two are equivalent
  • Testing for association between group and outcome, when there are more than 2 groups which are not in some solid order1 means comparing a summary of the response between \(k\) groups, sometimes in a pairwise fashion

1 The dose of a drug or the severity of pain are examples of ordered variables.

12.2 Covariable-Unadjusted Analyses

Appropriate when

  • Only interested in assessing the relationship between a single \(X\) and the response, or
  • Treatments are randomized and there are no strong prognostic factors that are measureable
  • Study is observational and variables capturing confounding are unavailable (place strong caveats in the paper)

See Chapter 13

12.2.1 Analyzing Paired Responses

Type of Response Recommended Test Most Frequent Test
binary McNemar McNemar
continuous Wilcoxon signed-rank paired \(t\)-test

12.2.2 Comparing Two Groups

Type of Response Recommended Test Most Frequent Test
binary \(2\times 2~\chi^{2}\) \(\chi^{2}\), Fisher’s exact test
ordinal Wilcoxon 2-sample Wilcoxon 2-sample
continuous Wilcoxon 2-sample 2-sample \(t\)-test
time to event2 Cox model3 log-rank4

2 The response variable may be right-censored, which happens if the subject ceased being followed before having the event. The value of the response variable, for example, for a subject followed 2 years without having the event is 2+.

3 If the treatment is expected to have more early effect with the effect lessening over time, an accelerated failure time model such as the lognormal model is recommended.

4 The log-rank is a special case of the Cox model. The Cox model provides slightly more accurate \(P\)-values than the \(\chi^2\) statistic from the log-rank test.

12.2.3 Comparing \(>2\) Groups

Type of Response Recommended Test Most Frequent Test
binary \(r\times 2~\chi^{2}\) \(\chi^{2}\), Fisher’s exact test
ordinal Kruskal-Wallis Kruskal-Wallis
continuous Kruskal-Wallis ANOVA
time to event Cox model log-rank

12.2.4 Correlating Two Continuous Variables

Recommended: Spearman \(\rho\)
Most frequently seen: Pearson \(r\)

12.3 Covariable-Adjusted Analyses

  • To adjust for imbalances in prognostic factors in an observational study or for strong patient heterogeneity in a randomized study
  • Analysis of covariance is preferred over stratification, especially if continuous adjustment variables are present or there are many adjustment variables
    • Continuous response: multiple linear regression with appropriate transformation of \(Y\)
    • Binary response: binary logistic regression model
    • Ordinal response: proportional odds ordinal logistic regression model
    • Time to event response, possibly right-censored:
      • chronic disease: Cox proportional hazards model
      • acute disease: accelerated failure time model