# 12 Statistical Inference Review

- Emphasize confidence limits, which can be computed from adjusted or unadjusted analyses, with or without taking into account multiple comparisons
- \(P\)-values can accompany CLs if formal hypothesis testing needed
- When possible construct \(P\)-values to be consistent with how CLs are computed

## 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 order
^{1}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 sec-ancova

### 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 event^{2} |
Cox model^{3} |
log-rank^{4} |

^{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