Statistics for clinical research An introductory course

Session 2 Comparing two groups

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

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

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

Proportions Proportion Proportion Binary outcome (e.g. yes/no) Binary outcome (e.g. yes/no) Number between 0 and 1 Number between 0 and 1 2x2 table 2x2 table Effect sizes Effect sizes Risk Difference (RD); Relative Risk (RR); Risk Difference (RD); Relative Risk (RR); Odds Ratio (OR) Group 1 Group 2 Positive p1p1p1p1 p2p2p2p2 Negative n1n1n1n1 n2n2n2n2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Forest Plot Plot effect sizes with confidence intervals Plot effect sizes with confidence intervals Useful in comparing multiple effect sizes Useful in comparing multiple effect sizes Go to applet on website: Go to applet on website:

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