Download presentation

Presentation is loading. Please wait.

Published byJustin Sills Modified over 3 years ago

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 (n 1,n 2 >20) 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 n 1,n 2 >20 –Non-normal otherwise (typically)

4
Large-Sample Test of Null hypothesis: The population means differ by 0 (which is typically 0): Alternative Hypotheses: –1-Sided: –2-Sided: Test Statistic:

5
Large-Sample Test of Decision Rule: –1-sided alternative If z obs z ==> Conclude If z obs Do not reject –2-sided alternative If z obs z ==> Conclude If z obs -z ==> Conclude If -z Do not reject

6
Large-Sample Test of Observed Significance Level (P-Value) –1-sided alternative P=P(z z obs ) (From the std. Normal distribution) –2-sided alternative P=2P( z |z obs | ) (From the std. Normal distribution) If P-Value then reject the null hypothesis

7
Large-Sample (1- 100% Confidence Interval for Confidence Coefficient (1- ) refers to the proportion of times this rule would provide an interval that contains the true parameter value if it were applied over all possible samples Rule:

8
Large-Sample (1- 100% Confidence Interval for For 95% Confidence Intervals, z.025 =1.96 Confidence Intervals and 2-sided tests give identical conclusions at same -level: –If entire interval is above 0, conclude –If entire interval is below 0, conclude –If interval contains 0, do not reject ≠

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 H 0 : 1 2 No difference in trt effects) H A : 1 2 ≠ Difference in trt effects) Test Statistic: Decision Rule ( =0.05) –Conclude > 0 since z obs = 25.3 > z.025 = 1.96

11
95% Confidence Interval for 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

12
Small-Sample Test for Normal Populations Case 1: Common Variances ( 1 2 = 2 2 = 2 ) Null Hypothesis : Alternative Hypotheses : –1-Sided: –2-Sided : Test Statistic: (where S p 2 is a “pooled” estimate of 2 )

13
Small-Sample Test for Normal Populations Decision Rule: (Based on t-distribution with =n 1 +n 2 -2 df) –1-sided alternative If t obs t , ==> Conclude If t obs Do not reject –2-sided alternative If t obs t , ==> Conclude If t obs -t ==> Conclude If -t Do not reject

14
Small-Sample Test for Normal Populations Observed Significance Level (P-Value) Special Tables Needed, Printed by Statistical Software Packages –1-sided alternative P=P(t t obs ) (From the t distribution) –2-sided alternative P=2P( t |t obs | ) (From the t distribution) If P-Value then reject the null hypothesis

15
Small-Sample (1- 100% Confidence Interval for Normal Populations Confidence Coefficient (1- ) refers to the proportion of times this rule would provide an interval that contains the true parameter value if it were applied over all possible samples Rule: Interpretations same as for large-sample CI’s

16
Small-Sample Inference for Normal Populations Case 2: 1 2 2 2 Don’t pool variances: Use “adjusted” degrees of freedom (Satterthwaites’ Approximation) :

17
Example - Scalp Wound Closure Groups: Stapling (n 1 =15) / Suturing (n 2 =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)

18
Example - Scalp Wound Closure H 0 : H A : 0 ( = 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 (n 1 +n 2 ). Ties take average ranks. –Obtain the rank sum for each group (T 1, T 2 ) –1-sided tests:Conclude H A : M 1 > M 2 if T 2 T 0 –2-sided tests:Conclude H A : M 1 M 2 if min(T 1, T 2 ) T 0 –Values of T 0 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 (n 1 = n 2 = 6) Outcome: Levocabastine AUC (1 Outlier/Group) 2-sided Test: Conclude Medians differ if min(T 1,T 2 ) 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: d i is the difference in scores (Trt 1 -Trt 2 ) for subject (pair) i Parameter: D - Population mean difference Sample Statistics:

23
Test Concerning D Null Hypothesis : H 0 : D = 0 (almost always 0) Alternative Hypotheses : –1-Sided: H A : D > 0 –2-Sided : H A : D 0 Test Statistic:

24
Test Concerning D Decision Rule: (Based on t-distribution with =n-1 df) 1-sided alternative If t obs t , ==> Conclude D If t obs Do not reject D 2-sided alternative If t obs t , ==> Conclude D If t obs -t ==> Conclude D If -t Do not reject D Confidence Interval for D

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 (1=Strongly Agree) Data: d i = 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 ( =0.05) H 0 : D = 0 H A : D 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 d i (as in the paired t-test) and obtain their absolute values (ignoring 0 s ) –Rank the observations by |d i | (smallest=1), averaging ranks for ties –Compute T + and T -, the rank sums for the positive and negative differences, respectively –1-sided tests:Conclude H A : M 1 > M 2 if T - T 0 –2-sided tests:Conclude H A : M 1 M 2 if min(T +, T - ) T 0 –Values of T 0 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 (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, =0.05, T 0 =2 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

Similar presentations

Presentation is loading. Please wait....

OK

T tests comparing two means t tests comparing two means.

T tests comparing two means t tests comparing two means.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on online voting system in php Download ppt on oxidation and reduction practice Download ppt on space exploration Ppt on effect of global warming on weather maps Ppt on flora and fauna of kerala Ppt on bullet train Ppt on dc motor drives Free pdf converter to ppt online Ppt on e commerce business models Ppt on review writing worksheets