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Statistical Comparison of Two Groups: Means & Proportions

Learn to compare two groups in clinical research, calculating effect sizes, risk differences, and more using Excel files provided.

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Statistical Comparison of Two Groups: Means & Proportions

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  1. Statistics for clinical research An introductory course

  2. Session 2 Comparing two groups

  3. Previous session • Normal distribution • Standard Deviation (of measurements) • Standard Error (of the mean) • Confidence Interval of measurements • Confidence Interval of the mean

  4. Main overview • Dealing with both Means and Proportions • Two groups will be compared • Effect Size along with its Confidence Interval(C.I.) will be calculated from data • Remember the C.I. tells us about the uncertainty of the effect size • The different calculations for effect sizes

  5. Means • Means calculated from measured data • Standard Deviation (of Measurements) • Standard Error (of the Mean) • Effect Size =Difference in Means

  6. Proportions • Proportion • Binary outcome (e.g. yes/no) • Number between 0 and 1 • 2x2 table • Effect sizes • Risk Difference (RD); Relative Risk (RR); Odds Ratio (OR)

  7. Comparing two groups Two proportions • Risk Difference • Number Needed to Treat • Relative Risk • Odds Ratio • Fisher’s Exact Probability Two means • The t-distribution • Difference between means

  8. Risk Difference • Risk is a proportion (number between 0 and 1) • Each group incorporate its own risk • Group 1: 15 people are given money… Happy = 12 Not happy = 3 Total = 15 Risk of happiness = 12/15 = 0.8 • Group 2: 10 people are not given money… Happy = 5 Not happy = 5 Total = 10 Risk of happiness = 5/10 = 0.5

  9. Risk Difference • Risk Difference (RD) is the risk of one group subtracted from the risk of the other group • RD = 0.8 – 0.5 = 0.3 • Excel file “TwoGroups.xls”

  10. Comparing two groups Two proportions • Risk Difference • Number Needed to Treat • Relative Risk • Odds Ratio • Fisher’s Exact Probability Two means • The t-distribution • Difference between means

  11. Number Needed to Treat • NNT = 1 / Risk Difference • If RD = 0.21 (21%), then need to treat 100 to prevent 21 adverse events • NNT = 1 / 0.21 = 5 (rounded up) • 5 need to be treated to prevent 1 additional adverse event • Excel file “TwoGroups.xls”

  12. Comparing two groups Two proportions • Risk Difference • Number Needed to Treat • Relative Risk • Odds Ratio • Fisher’s Exact Probability Two means • The t-distribution • Difference between means

  13. Relative Risk (RR) • Risk is a proportion • Each of the two groups has its own risk • Relative Risk (RR) is the ratio of two risks • RR is mostly used for cohort studies • Ratios do not have a Normal distribution • log(RR) has a Normal distribution • Confidence interval calculations require a Normal distribution • Excel file “TwoGroups.xls”

  14. Relative Risk (RR) • If Confidence Interval… • Contains 1: No difference in outcome between two groups • <1: Less risk in group 1 • >1: Greater risk in group 1

  15. Comparing two groups Two proportions • Risk Difference • Number Needed to Treat • Relative Risk • Odds Ratio • Fisher’s Exact Probability Two means • The t-distribution • Difference between means

  16. Odds Ratio (OR) • Odds – the number who have an event divided by the number who do not • Odds of an event occurring is obtained for both groups • OR mostly used for case-control studies • Ratios are not Normally distributed • log(OR) has a Normal distribution • Confidence Interval calculations require a Normal distribution • Extra: Logistic regression is typically used to adjust odds ratios to control for potential confounding by other variables • Excel file “TwoGroups.xls”

  17. Odds Ratio (OR) • If Confidence Interval… • Contains 1: No difference in outcome between two groups • <1: Odds in group 1 significantly less • >1: Odds in group 1 significantly greater

  18. Comparing two groups Two proportions • Risk Difference • Number Needed to Treat • Relative Risk • Odds Ratio • Fisher’s Exact Probability Two means • The t-distribution • Difference between means

  19. Fisher’s Exact Test • Determines if significant associations exist between group and outcome • Used when sample sizes are small • i.e. cell count < 5 in a 2x2 table • Alternative to the Chi-Square test • Test only provides a p-value (no C.I.) • Probability of observing a result more extreme than that observed

  20. Comparing two groups Two proportions • Risk Difference • Number Needed to Treat • Relative Risk • Odds Ratio • Fisher’s Exact Probability Two means • The t-distribution • Difference between means

  21. The t-distribution • Population SD is unknown and is estimated from the data • Blue curve = Normal distribution • Green = t-distribution with 1 degree of freedom (df) • Red = t-distribution, 2 df • Underlying theory of the t-test

  22. Comparing two groups Two proportions • Risk Difference • Number Needed to Treat • Relative Risk • Odds Ratio • Fisher’s Exact Probability Two means • The t-distribution • Difference between means

  23. Difference between means • Two sample t-test is used to test the difference between two means • Measurements must be considered Normally distributed • Quite powerful. A decision can be made with a small sample size…much smaller than when compared to proportions • Excel file “TwoGroups.xls”

  24. Forest Plot • Plot effect sizes with confidence intervals • Useful in comparing multiple effect sizes • Go to applet on website: http://www.materrsc.org/Course/CI_Diff.html

  25. Additional topics • Normality tests (e.g. Shapiro-Wilk) • Test for equality of variances (e.g. Bartlett’s test) • t-test for unequal variances • Paired t-test for dependent samples • Comparing more than two groups (e.g. one-way ANOVA) • Nonparametric tests

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