T s and F s. Statistical testing for means. FETP India

Transcription

1 T s and F s Statistical testing for means FETP India

2 Competency to be gained from this lecture Test the statistical significance of the difference between two means

3 Key elements Paired and unpaired data Paired t-test Unpaired t-test F test

4 Means Proportion Application of the concept of statistical testing Measures of association Paired and unpaired data

5 Statistical testing for means Means T-test for paired data T-test for unpaired data F-test to test the difference in variances Proportion Measures of association Paired and unpaired data

6 Comparing unpaired data Concept Comparing a bag of observations against another bag of observations Example The mean height of the children in one class versus versus the height of the children in another Paired and unpaired data

7 Comparing paired data Concept Comparing pairs of observations that are linked with each other Example Pre and post treatment values of a parameter in a group of subjects Paired and unpaired data

8 Paired and unpaired t-tests To test the difference between two sample means that are paired (e.g., before and after treatment) or matched (e.g., patients matched for age, sex, etc) Use PAIRED t-test To test the difference between two sample means that are not paired / unmatched Use UNPAIRED (independent) t-test Paired and unpaired data

9 Example Drug trial Drug A and Drug B Two groups have equal initial blood sugars levels Question: Does the drug have an impact on the blood sugar level? Null hypothesis There is no difference between the mean blood sugar levels before and after treatment Paired and unpaired data

10 Options available for the example considered Two paired t-tests Each group has an initial and a post treatment values Two paired t-tests are possible for each group This option is adapted to a research question examining the individual relevance of each drug One unpaired test test on final value This option is adapted to a research question comparing the two drugs Paired and unpaired data

11 Methods to calculate the paired t-test: Concept We test the probability that the difference between the paired data is equal to 0 Paired t-test

12 Methods to calculate the paired t-test: Formula (1/2) Number of pairs: n Value before Rx: a Value after treatment: b Difference: Mean (d):

13 Methods to calculate the paired t-test: Formula (2/2) Variance (d):

14 Illustration of an application of the t-test Drug No of patients Fasting blood sugar (mg%) Initial Final Decrease A * B * * Statistically Significant ( P < 0.05) Paired t-test

15 Numerical example of paired t - test Patient number Erythrocyte sedimentation rate - 1 hour (mm) Before Rx (a) After Rx (b) Difference (a b) = d Square of difference (d 2 ) 289 1,088 1, , Total ,652 Paired t-test

16 d = 256 ; n = 10 ; d = 256/10 = 25.6 d 2 = 7652 Variance (s 2 ) = 1 d n 1 ( d n 2 2 ) 1 (256) = = S s = = = d 25.6 s / n / 10 t = = = 7.33 with 9 d.f. 2 Paired t-test

17 Inference Calculated value of t= degrees of freedom (df) Tabulated value of t (df=9)(0.1%) = The value of t-cal exceeding the value of t-tab The treatment had a significant benefit in reducing the erythrocyte sedimentation rate (P < 0.001) The mean erythrocyte sedimentation rate after treatment (7.0 mm) is significantly lower than the mean pre-treatment ESR value (32.6 mm) Paired t-test

18 Methods to calculate the unpaired t-test: Concept The pooled variance is a weighted average of the two variances If the two sample sizes are equal, the pooled variance is the mean of the two variances The t-table is identical for unpaired and paired data Unpaired t-test

19 Methods to calculate the unpaired t-test: Formula Sample I Sample II Size n 1 n 2 Mean x 1 x 2 Variance s 2 1 s 2 2 To test the significance of the difference between the two sample means, calculate x1 x2 x x SE( x x2) 1 2 t = = 1 s 2 1 n 1 1 n 2 (n 1-1) s (n 2-1) s 2 2 where s 2 = (n 1-1) + (n 2-1) t follows a t distribution with (n 1 + n 2-2) df Unpaired t-test

20 Numerical example of unpaired t -test Comparing the 24-hour total energy expenditure among an obese and a lean group Null hypothesis: There is no difference between the mean energy expenditure between the two groups

21 t cal > t tab indicate that the mean energy expenditure in obese group (10.3) is significantly (P<0.001) higher than that of lean group (8.1) Unpaired t-test

22 Underlying assumptions of the unpaired t-test 1. The distributions of x1 and x2 are normal 2. The population variances of x1 and x2 are equal However, minor deviations from these assumptions do not affect the validity of the test Unpaired t-test

23 Un-paired t-test on paired data It would be inefficient to test paired observations as though they were unpaired Consequences: Underestimation of t - value Overestimation of probability value Undercalling of significant difference Unpaired t-test

24 Unequal variances Variances in the two samples may differ considerably from one another Example: Two technicians, one experienced (more consistent) and the other relatively inexperienced (more variable) undertake a blood count Both technicians are estimating the same population mean value The more experienced one will have a smaller variability in his readings than the less experienced one F test

25 Possible course of action for situations with unequal variances No course of action will suit all situations Options: Transform the values to some other scale (e.g. logarithmic) to equalize variances Use specific methods when this is not possible: Modified t test Fisher-Behren s test F test

26 Variance ratio test (F- test) To test the equality of two variances, s12 and s22, we use a statistical test called the variance ratio test (F-test) Calculate the ratio of the larger variance to the smaller variance s12 i.e., F = (s12 - larger variance) s22 F follows a F-distribution with (n1 1) and (n2 1) degrees of freedom F test

27 Example of variance ratio test (F-test) Variance in the infected group 10.9 (n1= 10) Variance in the control group 5.9 (n2 = 12) F is calculated as = 10.9 / 5.9 = ,11 degrees of freedom (n1 and n2-1) Tabulated F 9,11(5%) = 2.92 A calculated F (Fcal) smaller than the tabulated F (Ftab) indicates that the variances are equal F test

28 Assumptions of the variance ratio F test The two samples must be independent e.g., Two series of patients and not the same patients tested twice (before and after treatment) Both samples must have come from a normal distribution F test

29 What test should be used to test the difference between two means? Test the difference between two sample mean values Values are paired / matched Values are unpaired / unmatched Paired t-test Check if variances are equal Equal variances Different variances Unpaired t-test Modified t-test Fisher-Behren test

30 Key messages Determine whether the data are paired Used paired t-test for paired data Used unpaired t-test for unpaired data if the variances are comparable Test for the difference in variance with F- test and use other tests if variances differ