Comparison of 2 Population Means

1
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,n2gt20) vs Small

2
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)

3
Parameters/Estimates (Independent Samples)

• Parameter
• Estimator
• Estimated standard error
• Shape of sampling distribution
• Normal if data are normal
• Approximately normal if n1,n2gt20
• Non-normal otherwise (typically)

4
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

5
Large-Sample Test of m1-m2

• Decision Rule
• 1-sided alternative
• If zobs ? za gt Conclude m1-m2 gt D0
• If zobs lt za gt Do not reject m1-m2 D0
• 2-sided alternative
• If zobs ? za/2 gt Conclude m1-m2 gt D0
• If zobs ? -za/2 gt Conclude m1-m2 lt D0
• If -za/2 lt zobs lt za/2 gt Do not reject m1-m2
  D0

6
Large-Sample Test of m1-m2

• Observed Significance Level (P-Value)
• 1-sided alternative
• PP(z ? zobs) (From the std. Normal
  distribution)
• 2-sided alternative
• P2P( z? zobs ) (From the std. Normal
  distribution)
• If P-Value ? a, then reject the null hypothesis

7
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

8
Large-Sample (1-a)100 Confidence Interval for
m1-m2

• For 95 Confidence Intervals, z.0251.96
• Confidence Intervals and 2-sided tests give
  identical conclusions at same a-level
• If entire interval is above D0, conclude m1-m2 gt
  D0
• If entire interval is below D0, conclude m1-m2 lt
  D0
• If interval contains D0, do not reject m1-m2 ? D0

9
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)
10
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 (a0.05)
• Conclude m1-m2 gt 0 since zobs 25.3 gt z.025
  1.96

11
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 gt 0

12
Small-Sample Test for m1-m2 Normal 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)

13
Small-Sample Test for m1-m2 Normal Populations

• Decision Rule (Based on t-distribution with
  nn1n2-2 df)
• 1-sided alternative
• If tobs ? ta,n gt Conclude m1-m2 gt D0
• If tobs lt ta,n gt Do not reject m1-m2 D0
• 2-sided alternative
• If tobs ? ta/2 ,n gt Conclude m1-m2 gt D0
• If tobs ? -ta/2,n gt Conclude m1-m2 lt D0
• If -ta/2,n lt tobs lt ta/2,n gt Do not reject
  m1-m2 D0

14
Small-Sample Test for m1-m2 Normal Populations

• Observed Significance Level (P-Value)
• Special Tables Needed, Printed by Statistical
  Software Packages
• 1-sided alternative
• PP(t ? tobs) (From the tn distribution)
• 2-sided alternative
• P2P( t ? tobs ) (From the tn distribution)
• If P-Value ? a, then reject the null hypothesis

15
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 CIs

16
Small-Sample Inference for m1-m2 Normal
Populations

• Case 2 s12 ? s22
• Dont pool variances
• Use adjusted degrees of freedom
  (Satterthwaites Approximation)

17
Example - Scalp Wound Closure

• Groups Stapling (n115) / Suturing (n216)
• 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)
18
Example - Scalp Wound Closure
H0 m1-m2 0 HA m1-m2 ? 0 (a
0.05)
No significant difference between 2 methods
19
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 (n1n2). Ties take average ranks.
• Obtain the rank sum for each group (T1 , T2 )
• 1-sided testsConclude HA M1 gt M2 if T2 ? T0
• 2-sided testsConclude HA M1 ? M2 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.

20
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)
21
Computer Output - SPSS

22
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

23
Test Concerning mD

• Null Hypothesis H0mDD0 (almost always 0)
• Alternative Hypotheses
• 1-Sided HA mD gt D0
• 2-Sided HA mD ? D0
• Test Statistic

24
Test Concerning mD
Decision Rule (Based on t-distribution with
nn-1 df) 1-sided alternative If tobs ? ta,n
gt Conclude mD gt D0 If tobs lt ta,n gt Do
not reject mD D0 2-sided alternative If tobs ?
ta/2 ,n gt Conclude mD gt D0 If tobs ?
-ta/2,n gt Conclude mD lt D0 If -ta/2,n lt
tobs lt ta/2,n gt Do not reject mD D0
Confidence Interval for mD
25
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 (1Strongly Agree)
• Data di difference (O.C.-EVRA) for subject i
• Summary Statistics

Source Rubinstein, et al (2004)
26
Example - Evaluation of Transdermal Contraceptive
Patch In Adolescents

• 2-sided test for differences in ease of use
  (a0.05)
• H0mD 0 HAmD ? 0

Conclude Mean Scores are higher for O.C., girls
find the Patch easier to use (low scores are
better)
27
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 (smallest1),
  averaging ranks for ties
• Compute T and T-, the rank sums for the positive
  and negative differences, respectively
• 1-sided testsConclude HA M1 gt M2 if T- ? T0
• 2-sided testsConclude HA M1 ? M2 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.

28
Example - New MRI for 3D Coronary Angiography

• Previous vs new Magnetization Prep Schemes (n7)
• Response Blood/Myocardium Contrast-Noise-Ratio

• All Differences are negative, T- 127 28,
  T 0
• From tables for 2-sided tests, n7, a0.05,
  T02
• Since min(0,28) ? 2, Conclude the scheme means
  differ

Source Nguyen, et al (2004)
29
Computer Output - SPSS


Note that SPSS is taking NEW-PREVIOUS in top table