Statistics in medicine

Transcription

1 Statistics in medicine Lecture 3: Bivariate association : continuous variables Fatma Shebl, MD, MS, MPH, PhD Assistant Professor Chronic Disease Epidemiology Department Yale School of Public Health Fatma.shebl@yale.edu S L I D E 0

2 Outline Mean in one group: Normal distribution Mean in the same group repeated twice: Normal distribution Mean in one group: Not normal distribution Mean in the same group repeated twice: Not normal distribution Mean in two groups: Normal distribution Mean in two groups: Not normal distribution Mean in three or more groups: Normal distribution Mean in three or more groups : Not normal distribution S L I D E 1

3 Bivariate analysis Bivariate Onesample > Onesample Mean Proportio n Continuo us and categoric al variable Continuo us and continuo us variable Categoric al and categoric al variable Two categories >= two categories Z test McNemar Correlatio n ANOVA Kruskal- Wallis test Chisquare test Fisher s exact test T-test Paired t- test Sign Wilcoxon sign rank test Pooled t- test Welch s t-test Wilcoxon rank sum test S L I D E 2

4 Statistical tests of one group One group Mean Proportion Normal distribution Not normal distribution Single times Two times/paired Single time Two times/paired Transformation Nonparametric test Z test McNemar test T test Paired-t test Single time Two times/paired Sign test Wilcoxon sign rank test S L I D E 3

5 Mean in one group: Normal distribution Single time point: One sample Student s t-test Is a test to compare mean of a sample to a known value Can be used for small sample size (<30) Use the t-distribution Violations of assumptions of t-test Nonnormality Skeweness Outliers Dependence between units of the sample S L I D E 4

6 Mean in one group: Normal distribution Single time point: One sample Student s t-test E.g., is the mean energy intake of 2- year-olds (i.e. mean of the sample) significantly different from the value reported in NHANES III (i.e. known value)? S L I D E 5

7 Steps of the statistical significance testing, t-test 1- Calculate the test statistic (critical ratio) 2- Calculate degrees of freedom 3- Determine the critical value of significance 4-Compare the test statistic with the critical value 5- Calculate the p value 6- Calculate the confidence interval 7- Draw a conclusion S L I D E 6

8 Mean in one group: Normal distribution, t-test 1- Calculate the critical value Critical ratio (critical value) General formula= t= t= Parameter SE of that parameter observed mean population mean SE of the mean x μ SD n 2- Calculate degrees of freedom One sample t test df= N-1 S L I D E 7

9 Mean in one group: Normal distribution 3-Determine the critical value of significance Critical value: The value that a test statistic must exceed (in absolute value sense) for the null hypothesis to be rejected Depends on: Alpha level Degrees of freedom Two-tail vs. one-tail test E.g., for an alpha of.05, two-tailed test, 8 df, the value of the t that defines the central 95% area (acceptance area) is between and Levels of Significance for a One-Tailed Test df Levels of Signficance for a Two-Tailed Test Source: S L I D E 8

10 Mean in one group: Normal distribution 4-Compare the test statistic with the critical value The test statistic is < the critical value (in absolute sense) acceptance area The test statistic is > the critical value (in absolute sense) rejection area S L I D E 9

11 Mean in one group: Normal distribution 5- Calculate the p value From the t-table Locate the row of the relevant df Find the t value that is < test statistic Find the p value at the top of the column E.g., for a two-tailed test, 8 df, t= > p value >.01 Levels of Significance for a One-Tailed Test df Levels of Signficance for a Two-Tailed Test S L I D E 10

12 Mean in one group: Normal distribution 6- Calculate the confidence interval CI of mean in one group: x ± t SE E.g. if the test has 8 df 95% CI= x ± SE Obtain this value from the t table S L I D E 11

13 Mean in one group: Normal distribution 7- Draw a conclusion Reject or fail to reject the null hypothesis Fail to reject the null If the p value > alpha level If the confidence interval crosses the 0 Reject the null If the p value < alpha level If the confidence interval does not include 0 S L I D E 12

14 One-sample t-test example A study of 94 2-year old energy intake compared to NHANES III mean energy intake value were conducted. The calculated t value was What is the corresponding p value? Answer: df=94-1=93 0.1>P value>0.05 Conclusion: we fail to reject the null hypothesis i.e. the 2- year old intake is not significantly different from NHANES III Levels of Significance for a One-Tailed Test df Levels of Signficance for a Two-Tailed Test Source: S L I D E 13

15 Mean in one group, paired/matched: Normal distribution One group is measured twice (or matched design) Paired Student s t-test Is a test to compare the difference (change) in the mean of two paired (matched) groups. Can be used for designs where individuals characteristics are measured before and after intervention Use the t-distribution Violations of assumptions of t-test Nonnormality Skeweness Outliers Dependence between units of the sample S L I D E 14

16 Mean in one group, paired/matched: Normal distribution, paired t-test One group is measured twice (or matched design): Paired Student s t-test E.g., Is there a change in serum LDL levels after cholecystectomy? S L I D E 15

17 Steps of the statistical significance testing, paired t-test 1- Calculate the test statistic (critical ratio) 2- Calculate degrees of freedom 3- Determine the critical value of significance 4-Compare the test statistic with the critical value 5- Calculate the p value 6- Calculate the confidence interval 7- Draw a conclusion S L I D E 16

18 Mean in one group, paired/matched: Normal distribution, paired t-test 1- Calculate the critical value Critical ratio (critical value) t= d 0 SD d n Where, d = mean difference d d 2 SD d = n 1 2- Calculate degrees of freedom Paired-t test df= N-1 S L I D E 17

19 Mean in one group, paired/matched: Normal distribution, paired t-test 3-Determine the critical value of significance Critical value: Depends on: Alpha level Degrees of freedom Two-tail vs. one-tail test E.g., for an alpha of.01, two-tailed test, 50 df, the value of the t that defines the central 99% area (acceptance area) is between and Levels of Significance for a One-Tailed Test df Levels of Signficance for a Two-Tailed Test Source: S L I D E 18

20 Mean in one group, paired/matched: Normal distribution, paired t-test 4-Compare the test statistic with the critical value The test statistic is < the critical value (in absolute sense) acceptance area The test statistic is > the critical value (in absolute sense) rejection area S L I D E 19

21 Mean in one group, paired/matched: Normal distribution, paired t-test 5- Calculate the p value From the t-table Locate the row of the relevant df Find the t value that is < test statistic Find the p value at the top of the column E.g., for a two-tailed test, 50 df, t= > p value >.001 Levels of Significance for a One-Tailed Test df Levels of Signficance for a Two-Tailed Test S L I D E 20

22 Mean in one group, paired/matched: Normal distribution, paired t-test 6- Calculate the confidence interval CI of mean difference in paired design: d ± t n 1 SD d n E.g. if the test has 50 df 95% CI= d ± SD d n S L I D E 21

23 Mean in one group, paired/matched: Normal distribution, paired t-test 7- Draw a conclusion Reject or fail to reject the null hypothesis Fail to reject the null If the p value > alpha level If the confidence interval crosses the 0 Reject the null If the p value < alpha level If the confidence interval does not include the 0 S L I D E 22

24 Paired t-test example A study of LDL in 51 patients who had cholecystectomy were conducted. The calculated t value was What is the corresponding p value? Answer: df=51-1=50 P value<0.001 Conclusion: we reject the null hypothesis i.e. the post operative LDL is significantly different from the pre operative LDL Levels of Significance for a One-Tailed Test df Levels of Signficance for a Two-Tailed Test S L I D E 23

25 Median in one group: Not normal distribution Single time point: Sign test Is a nonparametric (distribution free) test to compare median of a sample to a known value Indications: single sample with Abnormal distribution Skewed data Outliers S L I D E 24

26 Median in one group: Not normal distribution Single time point: Sign test Because the median is used, binomial distribution with π=.5 could be used If sample size is large, z distribution could be used E.g., is the median energy intake of 2- year-olds significantly different from the value reported in NHANES III? S L I D E 25

27 Steps of the statistical significance testing, sign test 1- Calculate the test statistic 2- Determine the critical value of significance 3-Compare the test statistic with the critical value 4- Calculate the p value 5- Draw a conclusion S L I D E 26

28 Median in one group: Not normal distribution, sign test 1- Calculate the critical value Critical ratio (critical value) Binomial distribution: p(x)= n! x! n x! πx (1 π) n x Z approximation: z= X nπ (1 2 ) nπ(1 π) S L I D E 27

29 Median in one group: Not normal distribution, sign test 2-Determine the critical value of significance Critical value identification Find the area For +z one-tailed test: 1-alpha two-tailed test: 1- (alpha/2) For -z one-tailed test: alpha two-tailed test: (alpha/2) Look in the table for the area value and find the corresponding z critical value E.g., for an alpha of.05, two-tailed test, the value of the z that defines the central 95% area (acceptance area) is between and 1.96 Source of table: phpapp02/95/copy-of-z-table jpg?cb= S L I D E 28

30 Median in one group: Not normal distribution, sign test 3-Compare the test statistic with the critical value The test statistic is < the critical value (in absolute sense) acceptance area The test statistic is > the critical value (in absolute sense) rejection area S L I D E 29

31 Median in one group: Not normal distribution, sign test 4- Calculate the p value Using the z-table Find the z value that is =test statistic by scrolling down the z column then scrolling to the right in the row The entries of the tables are used to calculate the p values For negative z values One-tailed, the p value is the entry of the z table Two-tailed, the p value is the entry of the z table multiplied by 2 For positive z values One-tailed, the p value = 1- the entry of the z table Two-tailed,: p value = (1- the entry of the z table )multiplied by 2 S L I D E 30

32 Median in one group: Not normal distribution, sign test 4- Calculate the p value E.g., for a z=3.1 one-tailed test, p=(1-.999) =.001 two-tailed test, p=(1-.999) *2=.002 Source of table : S L I D E 31

33 Median in one group: N0t normal distribution, sign test 5- Draw a conclusion Reject or fail to reject the null hypothesis Fail to reject the null If the p value > alpha level Reject the null If the p value < alpha level S L I D E 32

34 One-sample sign test example A study of 94 2-year old energy intake compared to NHANES III median energy intake value were conducted. The calculated z value was Using a two-tail test and.05 alpha level, what is the critical value and the corresponding p value? Answer: Critical value=+1.96 p=( ) *2=.0548 Conclusion: we fail to reject the null hypothesis i.e. the 2- year old intake is not significantly different from NHANES III Source: S L I D E 33

35 Mean in one group, paired/matched: Not normal distribution One group is measured twice (or matched design) Sign test Used for the difference of means Wilcoxon signed rank test A non parametric test for two DEPENDENT samples with ordinal data or with numerical observations that are NOT NORMALLY distributed As powerful as t-test More powerful than sign test Requires extensive computation, therefore statistical programs are needed S L I D E 34

36 Statistical tests of two independent groups Two groups Mean Proportion Normal distribution Not normal distribution Z test Chisquare test Fisher s exact test Risk ratios Equal variance Unequal variance Transformation Nonparametric test Pooled t test Welch s t test Wilcoxon rank sum test S L I D E 35

37 Mean in two groups: Normal distribution Two-sample t-test Is a test to compare means of two small samples Can be used for small sample size (<30) Use the t-distribution Violations of assumptions of t-test Nonnormality in either group: if sample size>30 it might be okay Equal variance in the two groups Dependence between units of the sample S L I D E 36

38 Mean in two groups: Normal distribution, t-test Two-sample t-test: E.g., Did women who receive paracervical block prior to the cryosurgery have less severe total cramping than women who did not? S L I D E 37

39 Steps of the statistical significance testing, Two sample t-test 1- Determine the type of t test 2- Calculate the test statistic 3- Calculate degrees of freedom 4- Determine the critical value of significance 5-Compare the test statistic with the critical value 6- Calculate the p value 7- Calculate the confidence interval 8- Draw a conclusion S L I D E 38

40 Steps of the statistical significance testing, Two sample t-test 1- Determine the type of t test Examine the equality of the variance in the two groups F test for equal variance Sensitive for data normality i.e. may be significant because data is not normal, not because the unequal variance If significant reject the hypothesis of equal variance Levene test for equal variances If significant reject the hypothesis of equal variance S L I D E 39

41 Steps of the statistical significance testing, Two sample t-test 1- Determine the type of t test Equal variance in the two groups Pooled t test Unequal variance in the two groups Welch s t test S L I D E 40

42 Steps of the statistical significance testing, Two sample t-test 2- Calculate the critical value Equal variances: Pooled t test t n1+n2-2 = x 1 x 2 SD p [(1/n 1 )+(1/n 2 )] Where, pooled SD p = SD p = n 1 1 SD n 2 1 SD 2 2 n 1 + n 2 2 S L I D E 41

43 Steps of the statistical significance testing, Two sample t-test 2- Calculate the critical value Unequal variances: Welch s t test t= x 1 x 2 SD Where, SD = SD= SD SD 2 2 n 1 n 2 S L I D E 42

44 Steps of the statistical significance testing, Two sample t-test 3- Calculate degrees of freedom Pooled t test df= n 1 + n 2 2 Welch s t test df= ( SD 1 2 n1 +SD 2 2 n2 )2 SD 2 n2 )2 ( SD 1 2 n1 )2 n1 1 +( n2 1 S L I D E 43

45 Steps of the statistical significance testing, Two sample t-test 4-Determine the critical value of significance, using t-distribution table based on Alpha level df Two-tail or one-tail 5-Compare the test statistic with the critical value The test statistic is < the critical value (in absolute sense) acceptance area The test statistic is > the critical value (in absolute sense) rejection area S L I D E 44

46 Steps of the statistical significance testing, Two sample t-test 6- Calculate the p value from the t-distribution table 7- Calculate the confidence interval CI of mean difference between two groups: mean difference ± t 1 α, df SE x 1 x 2 Where, Equal variance: SE x 1 x 2 = 1 n n 2 Unequal variance: SE x 1 x 2 = SD 1 2 n 1 + SD 2 2 n 2 S L I D E 45

47 Steps of the statistical significance testing, Two sample t-test 8- Draw a conclusion Reject or fail to reject the null hypothesis Fail to reject the null If the p value > alpha level If the confidence interval crosses the 0 Reject the null If the p value < alpha level If the confidence interval does not include 0 S L I D E 46

48 Steps of the statistical significance testing, Two sample t-test, example A study of cramping score in two groups of women (group1: 45 who received paracervical block, and group2: 39 who did not receive paracervical block) who underwent cryosurgery were conducted. The mean and SD were (35.6,and 28.45), and (51.41, and 8.11) in group1 and group2 respectively. The calculated t value was Assuming an alpha level of 0.01, and the alternative hypothesis is that women who receive paracervical block prior to the cryosurgery have less severe total cramping than women who did not. Calculate df, critical value, and p value? Source: S L I D E 47

49 Steps of the statistical significance testing, Two sample t-test, example Answer: df= =82 Critical value= <P value <0.01 Conclusion: we reject the null hypothesis i.e. women who received the block had significantly lower total cramping than women who did not receive the block Levels of Significance for a One-Tailed Test df Levels of Signficance for a Two-Tailed Test Source: S L I D E 48

50 Mean in two groups: Not normal distribution, Wilcoxon rank sum test A nonparametric test for comparing two independent samples with ordinal data or with numerical observations that are not normally distributes It tests the hypothesis that the means of the ranks are equal Steps: 1.Rank or observations regardless of the group 2.Sum the ranks of each group 3.Calculate the mean and SD of the ranks in each group 4.Calculate pooled SD of the ranks S L I D E 49