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## Comparison of 2 Population Means

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**Comparison of 2 Population Means**• Goal: To compare 2 populations/treatments wrt a numeric outcome • Sampling Design: Independent Samples (Parallel Groups) vs Paired Samples (Crossover Design) • Data Structure: Normal vs Non-normal • Sample Sizes: Large (n1,n2>20) vs Small**Independent Samples**• Units in the two samples are different • Sample sizes may or may not be equal • Large-sample inference based on Normal Distribution (Central Limit Theorem) • Small-sample inference depends on distribution of individual outcomes (Normal vs non-Normal)**Parameters/Estimates (Independent Samples)**• Parameter: • Estimator: • Estimated standard error: • Shape of sampling distribution: • Normal if data are normal • Approximately normal if n1,n2>20 • Non-normal otherwise (typically)**Large-Sample Test of m1-m2**• Null hypothesis: The population means differ by D0 (which is typically 0): • Alternative Hypotheses: • 1-Sided: • 2-Sided: • Test Statistic:**Large-Sample Test of m1-m2**• Decision Rule: • 1-sided alternative • If zobsza ==> Conclude m1-m2 > D0 • If zobs < za ==> Do not reject m1-m2 = D0 • 2-sided alternative • If zobsza/2 ==> Conclude m1-m2 > D0 • If zobs -za/2 ==> Conclude m1-m2 < D0 • If -za/2 < zobs < za/2 ==> Do not reject m1-m2 = D0**Large-Sample Test of m1-m2**• Observed Significance Level (P-Value) • 1-sided alternative • P=P(z zobs) (From the std. Normal distribution) • 2-sided alternative • P=2P(z |zobs| )(From the std. Normal distribution) • If P-Value a, then reject the null hypothesis**Large-Sample (1-a)100% Confidence Interval for m1-m2**• Confidence Coefficient (1-a) refers to the proportion of times this rule would provide an interval that contains the true parameter value m1-m2 if it were applied over all possible samples • Rule:**Large-Sample (1-a)100% Confidence Interval for m1-m2**• For 95% Confidence Intervals, z.025=1.96 • Confidence Intervals and 2-sided tests give identical conclusions at same a-level: • If entire interval is above D0, conclude m1-m2 > D0 • If entire interval is below D0, conclude m1-m2 < D0 • If interval contains D0, do not reject m1-m2 ≠ D0**Example: Vitamin C for Common Cold**• Outcome: Number of Colds During Study Period for Each Student • Group 1: Given Placebo • Group 2: Given Ascorbic Acid (Vitamin C) Source: Pauling (1971)**2-Sided Test to Compare Groups**• H0: m1-m2= 0 (No difference in trt effects) • HA: m1-m2≠ 0 (Difference in trt effects) • Test Statistic: • Decision Rule (a=0.05) • Conclude m1-m2> 0 since zobs = 25.3 > z.025= 1.96**95% Confidence Interval for m1-m2**• Point Estimate: • Estimated Std. Error: • Critical Value: z.025 = 1.96 • 95% CI: 0.30 ± 1.96(0.0119) 0.30 ± 0.023 (0.277 , 0.323) Entire interval > 0**Small-Sample Test for m1-m2Normal Populations**• Case 1: Common Variances (s12 = s22 = s2) • Null Hypothesis: • Alternative Hypotheses: • 1-Sided: • 2-Sided: • Test Statistic:(where Sp2 is a “pooled” estimate of s2)**Small-Sample Test for m1-m2Normal Populations**• Decision Rule: (Based on t-distribution with n=n1+n2-2 df) • 1-sided alternative • If tobsta,n ==> Conclude m1-m2 > D0 • If tobs < ta,n ==> Do not reject m1-m2 = D0 • 2-sided alternative • If tobsta/2 ,n ==> Conclude m1-m2 > D0 • If tobs -ta/2,n ==> Conclude m1-m2 < D0 • If -ta/2,n < tobs < ta/2,n ==> Do not reject m1-m2 = D0**Small-Sample Test for m1-m2Normal Populations**• Observed Significance Level (P-Value) • Special Tables Needed, Printed by Statistical Software Packages • 1-sided alternative • P=P(t tobs) (From the tn distribution) • 2-sided alternative • P=2P(t |tobs| )(From the tn distribution) • If P-Value a, then reject the null hypothesis**Small-Sample (1-a)100% Confidence Interval for m1-m2 -**Normal Populations • Confidence Coefficient (1-a) refers to the proportion of times this rule would provide an interval that contains the true parameter value m1-m2 if it were applied over all possible samples • Rule: • Interpretations same as for large-sample CI’s**Small-Sample Inference for m1-m2Normal Populations**• Case 2: s12 s22 • Don’t pool variances: • Use “adjusted” degrees of freedom (Satterthwaites’ Approximation) :**Example - Scalp Wound Closure**• Groups: Stapling (n1=15) / Suturing (n2=16) • Outcome: Physician Reported VAS Score at 1-Year • Conduct a 2-sided test of whether mean scores differ • Construct a 95% Confidence Interval for true difference Source: Khan, et al (2002)**Example - Scalp Wound Closure**H0: m1-m2 = 0 HA: m1-m2 0 (a = 0.05) No significant difference between 2 methods**Small Sample Test to Compare Two Medians - Nonnormal**Populations • Two Independent Samples (Parallel Groups) • Procedure (Wilcoxon Rank-Sum Test): • Rank measurements across samples from smallest (1) to largest (n1+n2). Ties take average ranks. • Obtain the rank sum for each group (T1 , T2 ) • 1-sided tests:Conclude HA: M1 > M2 if T2 T0 • 2-sided tests:Conclude HA: M1M2 if min(T1, T2) T0 • Values of T0 are given in many texts for various sample sizes and significance levels. P-values printed by statistical software packages.**Example - Levocabostine in Renal Patients**• 2 Groups: Non-Dialysis/Hemodialysis (n1 = n2 = 6) • Outcome: Levocabastine AUC (1 Outlier/Group) 2-sided Test: Conclude Medians differ if min(T1,T2) 26 Source: Zagornik, et al (1993)**Inference Based on Paired Samples (Crossover Designs)**• Setting: Each treatment is applied to each subject or pair (preferably in random order) • Data: di is the difference in scores (Trt1-Trt2) for subject (pair) i • Parameter: mD - Population mean difference • Sample Statistics:**Test Concerning mD**• Null Hypothesis: H0:mD=D0 (almost always 0) • Alternative Hypotheses: • 1-Sided:HA: mD > D0 • 2-Sided: HA: mDD0 • Test Statistic:**Test Concerning mD**• Decision Rule: (Based on t-distribution with n=n-1 df) • 1-sided alternative • If tobsta,n ==> Conclude mD> D0 • If tobs < ta,n ==> Do not reject mD= D0 • 2-sided alternative • If tobsta/2 ,n ==> Conclude mD> D0 • If tobs -ta/2,n ==> Conclude mD< D0 • If -ta/2,n < tobs < ta/2,n ==> Do not reject mD= D0 Confidence Interval for mD**Example - Evaluation of Transdermal Contraceptive Patch In**Adolescents • Subjects:Adolescent Females on O.C. who then received Ortho Evra Patch • Response: 5-point scores on ease of use for each type of contraception (1=Strongly Agree) • Data: di = difference (O.C.-EVRA) for subject i • Summary Statistics: Source: Rubinstein, et al (2004)**Example - Evaluation of Transdermal Contraceptive Patch In**Adolescents • 2-sided test for differences in ease of use (a=0.05) • H0:mD = 0 HA:mD 0 Conclude Mean Scores are higher for O.C., girls find the Patch easier to use (low scores are better)**Small-Sample Test For Nonnormal Data**• Paired Samples (Crossover Design) • Procedure (Wilcoxon Signed-Rank Test) • Compute Differences di (as in the paired t-test) and obtain their absolute values (ignoring 0s) • Rank the observations by |di| (smallest=1), averaging ranks for ties • Compute T+ and T-, the rank sums for the positive and negative differences, respectively • 1-sided tests:Conclude HA: M1 > M2 if T- T0 • 2-sided tests:Conclude HA: M1M2 if min(T+, T-) T0 • Values of T0 are given in many texts for various sample sizes and significance levels. P-values printed by statistical software packages.**Example - New MRI for 3D Coronary Angiography**• Previous vs new Magnetization Prep Schemes (n=7) • Response: Blood/Myocardium Contrast-Noise-Ratio • All Differences are negative, T- = 1+2+…+7 = 28, T+ = 0 • From tables for 2-sided tests, n=7, a=0.05, T0=2 • Since min(0,28) 2, Conclude the scheme means differ Source: Nguyen, et al (2004)**Computer Output - SPSS**Note that SPSS is taking NEW-PREVIOUS in top table