Cochran Mantel Haenszel Test Model Overview

The Cochran Mantel Haenszel Test is a family of stratified chi-square procedures. It asks whether two categorical variables remain associated after data are divided into meaningful strata such as schools, age groups, centres, study sites or risk categories. Unlike a single unadjusted chi-square test of independence, the analysis preserves the stratification variable and combines evidence across its levels.

This article uses the row-mean-scores member of the family. The row variable is sex with two groups, the ordered column variable is the G3 final-grade band scored 1 through 4, and school is the stratification variable. The procedure compares the observed male ordinal-score total with the score total expected under conditional independence inside each school. It then combines the two observed-minus-expected differences and their variances.

The Exact Research Question

After controlling for school, does the ordered distribution of final-grade bands differ between female and male students? The Cochran Mantel Haenszel Test answers this conditional question without pooling GP and MS as though they were one undifferentiated population.

Why “Cochran–Mantel–Haenszel” Can Mean Different Statistics

The phrase Cochran Mantel Haenszel Test is often used as though it identifies one statistic. In practice, it can refer to a 2 × 2 × K common-odds-ratio test, a correlation statistic, a row-mean-scores statistic or a general-association statistic. The table shape, measurement scale and selected score system determine which member is appropriate. This distinction prevents the common error of reporting a Mantel–Haenszel odds ratio for an outcome with four categories.

Quick Answer: Cochran Mantel Haenszel Test Result

The Cochran Mantel Haenszel Test compares sex with a four-level ordered final-grade variable while stratifying by school. Grade bands receive scores 1, 2, 3 and 4. The male score total is below its null expectation in GP and MS, producing a statistically significant one-degree-of-freedom row-mean-scores test.

Table of Contents

1. What the test means
2. Research question and hypotheses
3. When to use the procedure
4. CMH statistic variants
5. Variables and grade-band coding
6. Formula and manual calculation
7. Observed and expected tables
8. Complete statistical results
9. Assumptions and diagnostics
10. Comparison with related methods
11. Calculator workflow
12. Python chart stories
13. Paired R chart interpretations
14. Software workflows
15. Expandable code
16. APA reporting
17. Common mistakes
18. Practice questions
19. Downloads
20. Related guides
21. Frequently asked questions
22. Conclusion

What Is the Cochran Mantel Haenszel Test?

The Cochran Mantel Haenszel Test is a stratified procedure for categorical data. “Stratified” means the observations are organised into subgroups defined before analysis. The method calculates association information within each stratum and then combines that information, so the final result is adjusted for the stratifying factor.

Suppose a simple cross-tabulation indicates that sex and final-grade category are associated. That overall pattern might reflect a true within-school difference, a difference in the mix of students attending each school, or both. The Cochran Mantel Haenszel Test separates those possibilities by retaining school as a control variable rather than discarding it.

Conditional Independence

The null hypothesis is often described as conditional independence. Sex and the ordered grade band are conditionally independent given school when, inside each school stratum, the grade-band distribution does not systematically shift between female and male students. Conditional independence is more specific than the marginal independence tested after all strata are collapsed.

Why Scores Are Used

The four grade bands have a natural order: low, medium, high and very high. The row-mean-scores Cochran Mantel Haenszel Test assigns scores 1 through 4, preserving order and concentrating the test into one degree of freedom. A general-association test would treat the categories as nominal and use more degrees of freedom. The score-based version is therefore targeted to a monotonic ordinal shift.

What the Test Does Not Establish

A significant Cochran Mantel Haenszel Test does not prove that sex causes grade differences, that every individual follows the group pattern, or that school is the only confounder. It demonstrates a conditional association under the chosen scoring and stratification scheme. Causal claims require a design that addresses selection, confounding and measurement, as discussed in the guides to sampling methods and sampling bias.

Research Question, Hypotheses and Data Design

Research Question

After controlling for school, is the ordinal G3 grade-band distribution the same for female and male students?

Null and Alternative Hypotheses

Why the Hypotheses Must Be Prespecified

The score order and group direction should be selected before viewing the result. Changing from female-minus-male to male-minus-female changes the sign of z but not the two-sided p-value. Changing the grade scores can change the statistic itself. The distinction between planned and data-driven choices is part of responsible null and alternative hypothesis formulation and reduces the risk of selective reporting.

When to Use the Cochran Mantel Haenszel Test

Decision Flow

The Cochran Mantel Haenszel Test is especially useful when a simple pooled table may be confounded. It is not a universal substitute for regression; it is a focused stratified test that works best when the number of strata and table dimensions remain interpretable.

Cochran Mantel Haenszel Test Variants and Their Meanings

Choosing the correct CMH statistic is part of the analysis, not a cosmetic software option. The same three-way table can support different null hypotheses depending on whether the row and column variables are nominal or ordinal.

Why the Common Odds Ratio Is Not Reported

A pooled odds ratio requires binary row and column variables. Here the G3 outcome has four ordered categories. Collapsing it to pass/fail would answer a different question and discard information. If a project genuinely uses several 2 × 2 strata, the common-odds-ratio result should be paired with the Breslow–Day test to assess whether stratum-specific odds ratios are sufficiently homogeneous.

Variables, Coding and Data Dictionary

Why the Grade Bands Must Be Transparent

The Cochran Mantel Haenszel Test uses the assigned numeric scores, so the cut points and scores belong in the public method description. Readers should be able to reproduce the category conversion. A frequency and relative-frequency table should also be inspected before testing to ensure that no coding error has created empty or implausible categories.

Sample Composition

Cochran Mantel Haenszel Test Formula and Manual Calculation

The row-mean-scores Cochran Mantel Haenszel Test builds one score total and one null variance for each stratum. The worked calculation uses male as the focal row and column scores 1, 2, 3 and 4.

Notation

Step 1: Observed Male Score Total

GP: 1(23) + 2(108) + 3(44) + 4(11) = 415. MS: 1(27) + 2(43) + 3(8) + 4(2) = 145.

Step 2: Overall Mean Score Within Each Stratum

GP mean score = 2.3498818. MS mean score = 1.9646018. The lower MS mean reflects a different overall school profile, which is why school should not be ignored.

Step 3: Expected Male Score Total

GP expected male score = 186 × 2.3498818 = 437.0780142. MS expected male score = 80 × 1.9646018 = 157.1681416.

Step 4: Observed Minus Expected

GP D = 415 − 437.0780142 = −22.0780142. MS D = 145 − 157.1681416 = −12.1681416. The combined difference is −34.2461558.

Step 5: Fixed-Margin Variance

GP variance = 56.3582377. MS variance = 34.8486212. Their sum is 91.2068589.

Step 6: Combined CMH Statistic

Step 7: Signed Standard-Normal Score

The squared z equals the one-degree-of-freedom CMH statistic. The two-sided chi-square probability is p = 0.0003359187. This is far below .05 and .01.

Observed Tables, Expected Counts and Ordinal Profiles

GP School: Observed Counts

GP School: Row Percentages

Within GP, the male low-grade share is more than three times the female low-grade share. Female students have higher percentages in the high and very-high bands. The male mean score is 2.2312 compared with 2.4430 for females.

MS School: Observed Counts

MS School: Row Percentages

Within MS, males again have more mass in the two lower bands and less mass in the two upper bands. The male mean score is 1.8125 compared with 2.0479 for females. The direction therefore agrees with GP even though the school-wide grade profiles differ greatly.

Expected Counts Under Conditional Independence

For a refresher on how joint counts become row, column and total percentages, see two-way tables and relative frequencies.

Complete Cochran Mantel Haenszel Test Results

Primary Result Table

Stratum Score Components

Mean-Score Differences

The similar negative differences in GP and MS make the combined direction easy to interpret. The test does not estimate a causal effect or a common odds ratio. It establishes that the ordered grade profiles differ after stratifying by school.

Comparison Statistics from SPSS Crosstabs

The school-specific Pearson tests do not tell the same story because they use three degrees of freedom and evaluate any shape difference. The row-mean-scores Cochran Mantel Haenszel Test targets a consistent ordered shift and combines it across schools. This is an example of why the scientific question should determine the statistic rather than selecting whichever p-value is smallest.

Cochran Mantel Haenszel Test Assumptions and Diagnostic Checks

The Cochran Mantel Haenszel Test is robust only when its design assumptions match the data. These conditions differ from the familiar expected-count checklist used for a single Pearson table, although cell sparsity still matters for the asymptotic approximation.

Expected-Count Review

The GP table has no expected count below 5. The MS table has one expected cell below 5, with a minimum expected count of approximately 4.25. This does not automatically invalidate the score test, but it should be disclosed. For many small or sparse strata, consider exact conditional methods, permutation procedures or a model suited to sparse ordinal outcomes. The broader chi-square assumptions guide explains why expected-count rules are diagnostic rather than ritual.

Homogeneity Is Variant-Specific

The current row-mean-scores analysis does not estimate a common odds ratio, so the Breslow–Day homogeneity-of-odds-ratios assumption is not its primary condition. When the analysis is 2 × 2 × K and a pooled odds ratio is reported, however, heterogeneity must be examined rather than assumed.

Sensitivity to Scores and Cut Points

Scores 1–4 imply equal spacing between adjacent grade bands. A sensitivity analysis can compare table scores with rank or ridit scores. Results that change sharply under plausible scoring systems should be reported cautiously. It is also useful to analyse original G3 as a quantitative outcome when that scale remains scientifically defensible.

Independence Versus Clustering

School is treated as a fixed stratifier, not as a random cluster with many school-level units. With many schools, repeated observations or class-level dependence, a multilevel or generalised estimating approach may be preferable. The generalized linear model framework offers model-based extensions.

Cochran Mantel Haenszel Test Versus Related Tests

Versus a Chi-Square Test of Independence

A standard chi-square test of independence can be run on the collapsed sex-by-grade table, but it ignores school. Separate tests within schools do not provide one adjusted conclusion. The Cochran Mantel Haenszel Test combines within-stratum information while retaining the control structure.

Versus a Test for Homogeneity

A chi-square test of homogeneity asks whether categorical distributions are equal across independently sampled populations. The mathematics may resemble independence testing, but the sampling interpretation differs. The Cochran Mantel Haenszel Test adds a stratification layer and can exploit order.

Versus Ordinal Logistic Regression

Ordinal logistic regression can include school, sex, interactions and additional covariates, and it estimates adjusted odds ratios under a proportional-odds assumption. The Cochran Mantel Haenszel Test is simpler and less model-dependent but does not provide individual predictions or a full coefficient model.

Versus Binary Exact Tests

Barnard’s exact test, Boschloo’s test and the binomial test answer binary questions. They are not substitutes for a four-category stratified ordinal analysis unless the research question genuinely justifies binary recoding.

Cochran Mantel Haenszel Test Calculator: Step-by-Step Workflow

A trustworthy Cochran Mantel Haenszel Test calculator must identify which CMH statistic it computes. Entering a 2 × 4 × 2 table into a calculator designed only for 2 × 2 × K odds ratios will return the wrong analysis or reject the table.

Required Inputs

Calculation Checklist

1. Enter the eight GP counts and eight MS counts.
2. Confirm female and male totals of 383 and 266.
3. Confirm stratum totals of 423 and 226.
4. Assign grade scores 1–4 in increasing order.
5. Calculate male observed score totals of 415 and 145.
6. Calculate expected male score totals of 437.0780 and 157.1681.
7. Calculate variance components of 56.3582 and 34.8486.
8. Combine differences and variances to obtain CMH = 12.8587.
9. Use one chi-square degree of freedom to obtain p = 0.0003359.
10. Retain the negative signed z to explain direction.

The general chi-square calculator is useful for ordinary tables, but this score-based stratified calculation requires the dedicated formulas shown above.

Python Cochran Mantel Haenszel Test Chart Interpretations

The six Python figures translate the score-test calculation into observed totals, expected totals, deviations, variances, stratum tables and the final inferential result. Every chart should be read with the exact values rather than as decoration.

Python Chart 1: Cochran Mantel Haenszel Test — Observed and Expected Male Ordinal Scores

Pattern

In both schools, the observed bar is below the expected bar. The gap is larger in GP because GP has more students and a larger information contribution.

Key Values

GP: observed 415 versus expected 437.078. MS: observed 145 versus expected 157.168.

Interpretation

The male ordinal score total is lower than expected under conditional independence in each stratum. This consistent direction drives the negative signed z.

Why It Matters

A combined result is credible because the two school-specific deviations do not cancel one another.

Next step: Verify that the focal row and category scores match the written direction statement.

Python Chart 2: Cochran Mantel Haenszel Test — Score Deviations by School

Pattern

Both bars extend below zero. GP has a deviation near −22.08 and MS has a deviation near −12.17.

Key Values

Combined deviation: −34.2462. GP share: 64.47% of the absolute combined difference. MS share: 35.53%.

Interpretation

Negative values mean the male group accumulated fewer high-category score points than expected from the fixed margins.

Why It Matters

The sign communicates direction, while the square of the combined standardized deviation produces the chi-square statistic.

Next step: Inspect every stratum for sign reversals before reporting one combined direction.

Python Chart 3: Cochran Mantel Haenszel Test — Variance Contributions

Pattern

GP contributes more variance information than MS, consistent with its larger sample and category distribution.

Key Values

GP variance: 56.3582. MS variance: 34.8486. Total: 91.2069.

Interpretation

Variance prevents a large raw score difference from being interpreted without considering its expected sampling fluctuation.

Why It Matters

The final statistic is not a simple sum of school chi-square values; it standardizes the combined score difference by combined variance.

Next step: Recalculate variance if scores, margins or category order change.

Python Chart 4: Cochran Mantel Haenszel Test — GP Ordinal Grade Table

Pattern

Female and male patterns differ primarily at the low end and, to a lesser extent, in high and very-high categories.

Key Values

Female: 9, 137, 68, 23. Male: 23, 108, 44, 11. Female n = 237; male n = 186.

Interpretation

Raw counts must be read with denominators. The low-grade percentages are 3.80% for females and 12.37% for males.

Why It Matters

The GP table contributes a clear downward male shift and the larger of the two variance components.

Next step: Pair the heatmap with row percentages and score means rather than judging colour intensity alone.

Python Chart 5: Cochran Mantel Haenszel Test — MS Ordinal Grade Table

Pattern

Within MS, males remain more concentrated in low and medium categories and less represented in high and very-high categories.

Key Values

Female: 41, 67, 28, 10. Male: 27, 43, 8, 2. Female n = 146; male n = 80.

Interpretation

The male mean score is 1.8125 versus 2.0479 for females, a difference of −0.2354 score units.

Why It Matters

A pooled table could confuse school composition with within-school sex differences. Stratification separates those sources.

Next step: Report school-specific profiles so readers can see the consistency behind the combined statistic.

Python Chart 6: Cochran Mantel Haenszel Test — Result Summary

Pattern

The CMH bar reaches 12.86, signed z is −3.586 and −log10(p) is approximately 3.474.

Key Values

CMH: 12.8587. z: −3.5859. p: 0.0003359.

Interpretation

The chi-square statistic is nonnegative because it squares the standardized difference. Direction is preserved separately by z.

Why It Matters

Reporting both values avoids the common error of trying to infer direction from chi-square alone.

Next step: Use χ² and p for inference, then use signed z and stratum profiles for direction.

R Cochran Mantel Haenszel Test Charts in Paired Rows

The R figures are arranged in three pairs. Each pair places a computational component beside the diagnostic needed to interpret it. This layout keeps the Cochran Mantel Haenszel Test narrative horizontal on desktop and collapses cleanly on mobile.

Pair 1 — Direction and Null Expectation

R Chart 1

Cochran Mantel Haenszel Test Score Difference

Both score differences are below zero: −22.0780 in GP and −12.1681 in MS. The matching signs support one combined directional statement rather than a result driven by cancellation.

Report check: The two bars must match the Python deviation values and the Excel Diagnostics sheet.

R Chart 2

Cochran Mantel Haenszel Test Observed Versus Expected

The observed series stays below the expected series in both schools. The absolute levels differ because GP is larger and has a higher overall grade-score profile.

Report check: Do not compare vertical levels across schools without considering their different sample sizes.

Pair 2 — Information and Sample Structure

R Chart 3

Cochran Mantel Haenszel Test Variance by Stratum

GP contributes 56.3582 variance units and MS contributes 34.8486. The denominator sums these values before the score deviation is standardized.

Report check: The combined variance should equal 91.2068589.

R Chart 4

Cochran Mantel Haenszel Test Stratum Size

GP contains about 65.18% of the sample and MS about 34.82%. Larger size contributes information, but variance also depends on row margins and the distribution across score categories.

Report check: Stratum totals must sum to the verified N of 649.

Pair 3 — Standardized Evidence and Final Summary

R Chart 5

Cochran Mantel Haenszel Test Signed Standard-Normal Score

The point at −3.5859 lies well below −1.96, visually confirming a two-sided p-value below .05. Its negative location indicates a downward male score shift under the chosen coding.

Report check: Squaring −3.585899 gives 12.858673, the CMH chi-square statistic.

R Chart 6

Cochran Mantel Haenszel Test Result Summary

The summary reproduces the same three values as Python and Excel. The transformed p-value bar is a visualization aid; formal reporting should use the original p-value.

Report check: Report p = .000336 or p < .001, not p = 3.474.

Cochran Mantel Haenszel Test in Python, R, SPSS, SAS and Excel

Expandable Cochran Mantel Haenszel Test Code

The following code blocks reproduce the worked values. They intentionally calculate the ordinal row-mean-scores statistic instead of calling a 2 × 2 common-odds-ratio routine.

import math
import pandas as pd
from scipy.stats import chi2

df = pd.read_csv("dataset.csv")
df["grade_score"] = pd.cut(
    df["G3"], bins=[-math.inf, 9, 13, 16, math.inf], labels=[1, 2, 3, 4]
).astype(int)

scores = [1, 2, 3, 4]
sum_difference = 0.0
sum_variance = 0.0
components = []

for school, d in df.groupby("school", sort=True):
    tab = pd.crosstab(d["sex"], d["grade_score"]).reindex(
        index=["F", "M"], columns=scores, fill_value=0
    )
    female = tab.loc["F"].to_numpy(float)
    male = tab.loc["M"].to_numpy(float)
    nf, nm = female.sum(), male.sum()
    n = nf + nm
    totals = female + male
    mean_score = (totals * scores).sum() / n
    observed_male = (male * scores).sum()
    expected_male = nm * mean_score
    ss = (totals * (pd.Series(scores).to_numpy() - mean_score) ** 2).sum()
    variance = (nf * nm / (n * (n - 1))) * ss
    difference = observed_male - expected_male
    sum_difference += difference
    sum_variance += variance
    components.append([school, n, observed_male, expected_male, difference, variance])

z = sum_difference / math.sqrt(sum_variance)
cmh = z ** 2
p_value = chi2.sf(cmh, df=1)

print(components)
print(f"CMH={cmh:.12f}, z={z:.12f}, p={p_value:.12g}")

dat <- read.csv("dataset.csv", stringsAsFactors = FALSE)
dat$grade_score <- ifelse(dat$G3 < 10, 1,
                    ifelse(dat$G3 <= 13, 2,
                    ifelse(dat$G3 <= 16, 3, 4)))

scores <- 1:4
sum_diff <- 0
sum_var <- 0
components <- list()

for (school in sort(unique(dat$school))) {
  d <- dat[dat$school == school, ]
  tab <- table(factor(d$sex, levels=c("F","M")),
               factor(d$grade_score, levels=1:4))
  female <- as.numeric(tab["F", ])
  male <- as.numeric(tab["M", ])
  nf <- sum(female); nm <- sum(male); n <- nf + nm
  totals <- female + male
  mean_score <- sum(totals * scores) / n
  observed_male <- sum(male * scores)
  expected_male <- nm * mean_score
  ss <- sum(totals * (scores - mean_score)^2)
  variance <- (nf * nm / (n * (n - 1))) * ss
  difference <- observed_male - expected_male
  sum_diff <- sum_diff + difference
  sum_var <- sum_var + variance
  components[[school]] <- c(n=n, observed=observed_male,
                            expected=expected_male, difference=difference,
                            variance=variance)
}

z <- sum_diff / sqrt(sum_var)
cmh <- z^2
p_value <- pchisq(cmh, df=1, lower.tail=FALSE)
print(components)
cat(sprintf("CMH=%.12f, z=%.12f, p=%.12g\n", cmh, z, p_value))

DATASET CLOSE ALL.
OUTPUT CLOSE ALL.
GET DATA /TYPE=TXT
  /FILE='dataset.csv'
  /DELCASE=LINE /DELIMITERS="," /ARRANGEMENT=DELIMITED
  /FIRSTCASE=2 /IMPORTCASE=ALL.

* Recode strings to numeric variables as appropriate for your imported schema.
RECODE sex ('F'=0) ('M'=1) INTO sex_bin.
RECODE school ('GP'=1) ('MS'=2) INTO school_n.
COMPUTE grade3 = 1.
IF (G3 >= 10 AND G3 <= 13) grade3 = 2.
IF (G3 >= 14 AND G3 <= 16) grade3 = 3.
IF (G3 >= 17) grade3 = 4.
VALUE LABELS sex_bin 0 'Female' 1 'Male'
 /school_n 1 'GP' 2 'MS'
 /grade3 1 'Low below 10' 2 'Medium 10-13' 3 'High 14-16' 4 'Very high 17+'.
EXECUTE.

CROSSTABS
 /TABLES=sex_bin BY grade3 BY school_n
 /FORMAT=AVALUE TABLES
 /STATISTICS=CMH CHISQ
 /CELLS=COUNT ROW COLUMN EXPECTED
 /COUNT ROUND CELL.

data cmh_counts;
  length school $2 sex $1;
  input school $ sex $ grade3 count;
  datalines;
GP F 1 9
GP F 2 137
GP F 3 68
GP F 4 23
GP M 1 23
GP M 2 108
GP M 3 44
GP M 4 11
MS F 1 41
MS F 2 67
MS F 3 28
MS F 4 10
MS M 1 27
MS M 2 43
MS M 3 8
MS M 4 2
;
run;

proc freq data=cmh_counts order=data;
  tables school*sex*grade3 / cmh scores=table;
  weight count;
run;

Grade score: =IF(C5<10,1,IF(C5<=13,2,IF(C5<=16,3,4)))
Observed cell: =COUNTIFS(Data_Input!$B:$B,$A5,Data_Input!$A:$A,B$4,Data_Input!$D:$D,C$4)
Stratum mean score: =SUMPRODUCT(CategoryTotals,Scores)/StratumN
Expected male score: =MaleN*StratumMeanScore
Difference: =ObservedMaleScore-ExpectedMaleScore
CMH statistic: =(SUM(Differences)^2)/SUM(Variances)
P-value: =CHISQ.DIST.RT(CMH_Cell,1)
Signed z: =SUM(Differences)/SQRT(SUM(Variances))

How to Interpret and Report the Cochran Mantel Haenszel Test

APA-Style Worked Report

Result With Additional Descriptive Context

Reusable Reporting Templates

What to Include in a Complete Report

• The exact CMH variant and score system.
• The row, column and stratification variables.
• Observed counts or row percentages by stratum.
• CMH statistic, df, p-value and signed z.
• Direction convention and stratum-specific score differences.
• Any sparse-cell or scoring sensitivity note.
• A clear distinction between statistical association and causation.

A p-value should be accompanied by magnitude and context. The general effect-size guide explains why statistical significance alone is not a practical conclusion.

Common Cochran Mantel Haenszel Test Mistakes and Corrections

Threshold Thinking

The result is well below .05, but good reporting still avoids the word “proven.” A formal decision controls long-run error rates; it does not remove uncertainty or design limitations. Review Type I and Type II errors and statistical power when planning a new stratified study.

Cochran Mantel Haenszel Test Practice Questions

Use these questions to test whether the table design, score statistic and interpretation are understood rather than memorised.

Cochran Mantel Haenszel Test Reports and Worked Files

Each download reproduces the same variables, grade-band coding and primary result. The links include the download attribute so browsers can save the file instead of opening an embedded viewer when supported.

Related Salar Cafe Guides

Cochran Mantel Haenszel Test Frequently Asked Questions

Cochran Mantel Haenszel Test Conclusion

The Cochran Mantel Haenszel Test is the correct family of methods when categorical association must be evaluated within strata rather than after collapsing them. In this analysis, the ordinal row-mean-scores version compares sex with four ordered G3 grade bands while controlling for school.

The verified result is χ²(1) = 12.858673, p = 0.000335919, with signed z = −3.585899. Male observed score totals fall below their null expectations in GP and MS, and male mean grade-band scores are lower in both strata. The evidence therefore supports a conditional ordinal association, not a common odds ratio and not a causal claim.

A defensible report names the exact CMH variant, explains the score coding, provides stratum-specific counts, reports both chi-square and signed z, checks sparse cells and distinguishes the result from ordinary chi-square independence and model-based alternatives. That combination makes the Cochran Mantel Haenszel Test reproducible, interpretable and aligned with the actual research design.