Menna Arafat

6 Independent two-group example: biomarker change

6.1 Define the estimand

For Control and Drug_A, define biomarker change as

\[ \Delta B=B_{12}-B_0. \]

The target is the difference between the groups in mean biomarker change.

6.2 Descriptive statistics

ind_bio <- dat |> filter(treatment_arm %in% c("Control","Drug_A"), !is.na(biomarker_change)) |> droplevels()

ind_sum <- ind_bio |> group_by(treatment_arm) |>
                      summarise(
                      n=n(),
                      mean=mean(biomarker_change), sd=sd(biomarker_change),
                      median=median(biomarker_change), IQR=IQR(biomarker_change), .groups="drop")

kbl(ind_sum, digits=2, caption="Biomarker change in Control and Drug_A")
Table 6.1: Biomarker change in Control and Drug_A
treatment_arm n mean sd median IQR
Control 118 -2.71 9.68 -4.25 12.75
Drug_A 116 -14.94 11.93 -14.25 16.03

The mean and SD summarize the outcome on the scale relevant to mean comparisons. The median and IQR provide additional information about asymmetry and spread.

6.3 Distribution and influential observations

ggplot(ind_bio, aes(treatment_arm, biomarker_change)) +
  geom_boxplot(width=.55, outlier.shape=NA) +
  geom_jitter(width=.08, alpha=.30, size=1.3) +
  labs(x="Treatment group", y="Biomarker change")
Distribution of biomarker change in Control and Drug_A.

Figure 6.1: Distribution of biomarker change in Control and Drug_A.

The plot shows group location, spread, asymmetry, and individual observations. Large differences in spread are relevant to the pooled Student test, while extreme observations may affect any mean-based analysis.

6.4 Normality assessment

Normality is assessed separately within each independent group.

6.4.1 Q–Q plots

ggplot(ind_bio, aes(sample=biomarker_change)) +
  stat_qq(alpha=.45) + 
  stat_qq_line() + 
  facet_wrap(~treatment_arm) +
  labs(x="Theoretical normal quantiles", y="Observed quantiles")
Q–Q plots of biomarker change by treatment group.

Figure 6.2: Q–Q plots of biomarker change by treatment group.

Approximate alignment with the reference line supports a normal model. Systematic curvature suggests skewness, while marked tail departures may indicate heavy tails or influential observations.

6.4.2 Shapiro–Wilk tests

shapiro_bio <- ind_bio |> group_by(treatment_arm) |> group_modify(~{
  z <- shapiro.test(.x$biomarker_change)
  tibble(W=unname(z$statistic), p_value=z$p.value)
})

kbl(shapiro_bio, digits=3, caption="Shapiro–Wilk tests for biomarker change")
Table 6.2: Shapiro–Wilk tests for biomarker change
treatment_arm W p_value
Control 0.986 0.256
Drug_A 0.992 0.757

The Shapiro–Wilk test evaluates

\[ H_0:\text{the observations are compatible with a normal distribution}. \]

A small Shapiro–Wilk p-value indicates departure from normality, but test choice should not depend on it alone. With a large sample and only mild Q–Q plot deviations, a t-test is usually still appropriate. For independent groups, use Welch’s t-test if variances differ. If non-normality is substantial, especially with strong skewness or influential outliers, consider the Wilcoxon–Mann–Whitney test. For paired data, assess the paired differences and use either a paired t-test or Wilcoxon signed-rank test accordingly.

6.5 Equality of variance

A median-centered Levene test, often called the Brown–Forsythe test, assesses whether the group spreads differ.

lev_bio <- car::leveneTest(biomarker_change~treatment_arm, data=ind_bio, center=median)
lev_bio
#> Levene's Test for Homogeneity of Variance (center = median)
#>        Df F value Pr(>F)  
#> group   1    5.52   0.02 *
#>       232                 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The null hypothesis is

\[ H_0:\sigma_1^2=\sigma_2^2. \]

Evidence of unequal variances argues against the pooled Student test. Welch’s test does not require equal population variances.

6.6 Automated test selection

The following block combines the diagnostic results and performs one selected analysis.

normal_ok <- all(shapiro_bio$p_value>=.05)
levene_p <- as.numeric(lev_bio[1,"Pr(>F)"])
equal_var <- levene_p>=.05

if(normal_ok && equal_var){
  selected_test <- "Pooled Student t-test"
  fit <- t.test(biomarker_change~treatment_arm, data=ind_bio, var.equal=TRUE)
  reason <- "Normality acceptable; no evidence of unequal variances."
} else if(normal_ok){
  selected_test <- "Welch two-sample t-test"
  fit <- t.test(biomarker_change~treatment_arm, data=ind_bio, var.equal=FALSE)
  reason <- "Normality acceptable; variances differ."
} else {
  selected_test <- "Wilcoxon–Mann–Whitney rank-sum test"
  fit <- wilcox.test(biomarker_change~treatment_arm, data=ind_bio, exact=FALSE, conf.int=TRUE)
  reason <- "Evidence of departure from normality; rank-based analysis selected."
}

tibble(selected_test, reason) |> kbl(caption="Automated test selection")
Table 6.3: Automated test selection
selected_test reason
Welch two-sample t-test Normality acceptable; variances differ. 
broom::tidy(fit) |> kbl(digits=3, caption="Selected independent-group analysis")
Table 6.3: Selected independent-group analysis
estimate estimate1 estimate2 statistic p.value parameter conf.low conf.high method alternative
12.23 -2.712 -14.94 8.6 0 221 9.426 15.03 Welch Two Sample t-test two.sided

The operational rule is

Normality Variance assessment Selected analysis
acceptable similar pooled Student t-test
acceptable unequal Welch t-test
substantial departure either Wilcoxon–Mann–Whitney test

This is the general rule, however the threshold \(p=0.05\) should not be interpreted as a universal boundary. Q–Q plots, influential observations, sample size, and the intended estimand remain relevant.

6.7 Mean-based tests

For the pooled Student test,

\[ t= \frac{\bar y_1-\bar y_2} {s_p\sqrt{1/n_1+1/n_2}}, \]

where \(s_p\) is the pooled estimate of the common SD.

Welch’s test instead uses

\[ SE(\bar y_1-\bar y_2) = \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, \]

and estimates the degrees of freedom using the Welch–Satterthwaite approximation.

Both procedures estimate a difference in means. Their distinction concerns how uncertainty is estimated when group variances differ.

6.8 Wilcoxon–Mann–Whitney test

The Wilcoxon rank-sum, or Mann–Whitney, procedure replaces the original measurements by their ranks and evaluates the relative ordering of observations from the two groups.

If \(R_1\) is the rank sum for group 1,

\[ U_1=R_1-\frac{n_1(n_1+1)}{2}. \]

It should not automatically be described as a test of equal medians. A simple median or location-shift interpretation requires additional distributional conditions.

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