Below is a comprehensive statistical recommendation based on your study description --- # ### 1️⃣ SCENARIO CLASSIFICATION | Dimension | Classification | Explanation | | ------------------------------- | ------------------------------------------------------------------------------------ | ------------------------------------------------------------- | | **Research question type** | Difference | Comparing mean change in anxiety between two groups | | **Independent variables (IVs)** | 1 — Treatment condition (CBT vs. TAU) | Between-subjects categorical factor | | **Dependent variables (DVs)** | 1 — Change in GAD-7 score | Continuous outcome (difference score) | | **Measurement scales** | IV: Nominal (group labels); DV: Interval (summed Likert scale treated as continuous) | GAD-7 is commonly analyzed as continuous in clinical research | | **Number of groups** | 2 (CBT vs. TAU) | Classic two-arm RCT | | **Sample relationship** | Independent groups | Different participants in each arm | | **Sample sizes** | CBT: n = 45; TAU: n = 42 | Both > 30 → favorable for parametric tests | | **Distribution shape** | Approximately normal (slight skew in CBT) | No major violations reported | **Summary narrative:** This is a two-group, between-subjects comparison of a continuous outcome (change score). With reasonably large and balanced samples and approximate normality, the design aligns well with parametric mean-comparison tests. The independence of groups and single DV strongly constrain the choice toward a two-sample comparison framework. --- # ### 2️⃣ PRIMARY TEST RECOMMENDATION ## ✅ **Independent Samples t-test (Two-Sample t-test)** * **Test family:** Parametric, frequentist * **When to use:** * One categorical IV with 2 independent groups * One continuous DV * Approximately normal distributions * Homogeneity of variance (or use Welch variant if violated) ### **Plain-language rationale** You are comparing the *average change in anxiety scores* between two independent groups (CBT vs. TAU). The outcome is continuous and roughly normally distributed, and your sample sizes are sufficiently large to support parametric assumptions. The independent samples t-test is specifically designed to detect whether the difference between two group means is larger than what we would expect from random variation alone. Because participants are randomized and groups are independent, this test directly aligns with your RCT design. ### **What this test actually does** The t-test calculates the difference between the two group means and divides it by the variability in the data (standard error). It then evaluates whether this standardized difference is large enough to be unlikely under the null hypothesis (no true difference between groups). > 🔁 **Important note:** If homogeneity of variance is violated → use **Welch’s t-test**, which is more robust. --- # ### 3️⃣ ASSUMPTIONS & DIAGNOSTIC CHECKLIST | Assumption | How to Check | What Happens If Violated | Remediation | | | | ---------------------------------------- | ----------------------------------------------- | ---------------------------------------------------- | ---------------------------------------------- | -------------------------- | ----------------------------- | | **Independence of observations** | Study design (random assignment, no clustering) | Inflated Type I error | Use mixed models if clustering exists | | | | **Normality of DV within groups** | Shapiro-Wilk (p > .05), Q-Q plots | Reduced power or inaccurate p-values (small samples) | Use Mann-Whitney U or bootstrap CI | | | | **Homogeneity of variance** | Levene’s test (p > .05) | Biased standard errors, incorrect p-values | Use Welch’s t-test | | | | **Scale of measurement (continuous DV)** | Conceptual (GAD-7 treated as continuous) | Invalid test application | Use ordinal methods if strongly non-continuous | | | | **No extreme outliers** | Boxplots, z-scores (> | 3.29 | ) | Distorts mean and variance | Winsorize or use robust tests | --- ### ⚠️ Critical Violations * **Homogeneity of variance** is commonly violated → always check Levene’s test * **Outliers** can strongly distort results in clinical change scores → visually inspect carefully Ignoring these can lead to **false positives or misleading effect sizes**. --- # ### 4️⃣ ALTERNATIVE TESTS (RANKED) ## 1️⃣ **Welch’s t-test (Most common alternative)** * **Family:** Parametric * **Use instead if:** Variances are unequal (Levene’s p < .05) * **Trade-off:** Slightly less power but much more robust * **Decision rule:** 👉 IF variance homogeneity is violated → USE Welch’s t-test --- ## 2️⃣ **Mann–Whitney U test (Wilcoxon rank-sum)** * **Family:** Non-parametric * **Use instead if:** Strong non-normality or ordinal interpretation * **Trade-off:** * ✔ Robust to skew/outliers * ❌ Tests medians/ranks, not means * **Decision rule:** 👉 IF normality is violated AND data are skewed → USE Mann-Whitney U --- ## 3️⃣ **ANCOVA (Recommended in many RCTs)** * **Family:** Parametric (GLM) * **Use instead if:** You want to control for baseline GAD-7 instead of using change scores * **Trade-off:** * ✔ More statistical power * ✔ Adjusts for baseline imbalance * ❌ Slightly more complex * **Decision rule:** 👉 IF baseline scores vary meaningfully → USE ANCOVA (post-score as DV, baseline as covariate) --- # ### 5️⃣ EFFECT SIZE & POWER GUIDANCE ### **Effect size:** ✅ **Cohen’s d** * Represents standardized mean difference between groups * Example: * d = 0.5 → CBT group improved by half a standard deviation more than TAU ### **Benchmarks (rough guidelines)** * Small: 0.2 * Medium: 0.5 * Large: 0.8 ⚠️ Clinical context matters — even small effects may be meaningful in mental health --- ### **Sample size guidance** * For **medium effect (d = 0.5)**, α = .05, power = .80 👉 ~64 participants per group (you have slightly fewer, so power may be ~70–75%) --- ### **What to report** * t statistic * Degrees of freedom * p-value * Group means & SDs * 95% confidence interval * Cohen’s d --- # ### 6️⃣ IMPLEMENTATION QUICK-START ## **R** ```r # Standard t-test t.test(change ~ group, data = df, var.equal = TRUE) # Welch version (default) t.test(change ~ group, data = df) # Effect size library(effsize) cohen.d(change ~ group, data = df) ``` --- ## **Python** ```python from scipy import stats # Independent t-test stats.ttest_ind(cbt, tau, equal_var=True) # Welch's t-test stats.ttest_ind(cbt, tau, equal_var=False) ``` --- ## **SPSS** Analyze → Compare Means → Independent-Samples T Test * Move DV (change score) → Test Variable(s) * Move group → Grouping Variable * Define groups (e.g., 1 = CBT, 2 = TAU) --- ### **Reporting template** “A **independent samples t-test** revealed a **significant difference** in anxiety reduction between groups, *t*(df) = X.XX, p = .XXX, Cohen’s d = X.XX, indicating that the CBT group showed greater improvement than the TAU group.” --- # ### 7️⃣ COMMON PITFALLS & PRO TIPS ⚠️ **1. Using change scores without considering ANCOVA** → ANCOVA is often more powerful and preferred in RCTs ⚠️ **2. Ignoring variance inequality** → Always check Levene’s test; default to Welch if unsure ⚠️ **3. Overinterpreting p-values** → Statistical significance ≠ clinical significance ⚠️ **4. Not reporting effect sizes** → Always include Cohen’s d for interpretability ⚠️ **5. Assuming normality without checking** → Always inspect Q-Q plots, not just tests --- ## 💡 Pro Tip In clinical trials like yours, many statisticians **prefer ANCOVA over change scores** because it: * Adjusts for baseline differences * Provides greater statistical power * Reduces measurement error --- If you want, I can also show you how results might differ between **t-test vs. ANCOVA** on simulated data—this is often very insightful for RCT reporting.