Analysis of numerical data S4

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1 Basic medical statistics for clinical and experimental research Analysis of numerical data S4 Katarzyna Jóźwiak 3rd November /42

2 Hypothesis tests: numerical and ordinal data 1 group: one-sample t-test, sign test. 2 groups: Paired: paired t-test, Wilcoxon signed rank test, sign test Independent: unpaired t-test, Wilcoxon rank sum test. More than 2 groups: Paired: repeated measures ANOVA, Friedman s ANOVA Independent: one-way ANOVA, Kruskal-Wallis test. 2/42

3 Single group One-sample t-test (parametric): H 0 : population mean = parameter value H 1 : population mean parameter value (or H 1 : population mean > parameter value, or H 1 : population mean < parameter value) Example: sample from a single group of individuals (high school students) with their standardized test scores in writing. H 0 : population mean writing score = 5 H 1 : population mean writing score 5 Reject H 0 if: sample mean-parameter value sample mean-parameter value < c or > c SEM SEM with c > 0. 3/42

4 Single group Variable (e.g., writing test score) is numerical, is normally distributed with a given (usually unknown) variance. If population variance unknown: test statistic (here based on x 5) follows a Student-t distribution: 5% significance level: SEM = s/ n and c = tn 1,1 α/2 = t n 1,0.975 (s =standard deviation of sample observations, n =sample size) If population variance known (σ 2 ) or sample size very large: test statistic follows a normal distribution (z-test): 5% significance level: SEM = σ/ n and c = z1 α/2 = z = /42

5 Two related groups Paired data (matched pairs): One group of individuals and variable measured on each individual in two circumstances. E.g. measurement while taking active treatment and while taking placebo; blood pressure measured before and after a particular treatment. Two samples of different individuals but linked to each other. E.g. matched patients in case-control study Example: sample of patients in which hours of sleep under a sleeping drug is measured one night, and hours of sleep under a placebo is measured a different night for each. 5/42

6 Two related groups Paired t-test (parametric): H 0 : population mean group 1 - population mean group 2 = parameter value For the hours of sleep example: H 0 : pop. mean hours sleep (drug) - pop. mean hours sleep (placebo) = 0 H 1 : pop. mean hours sleep (drug) - pop. mean hours sleep (placebo) 0 Since data are paired, we reduce the two samples to a single sample of differences: Difference in hours sleep = hours sleep under drug - hours sleep under placebo Variable (difference in hours sleep) is numerical, is normally distributed with a given (usually unknown) variance. Hypotheses become: H 0 : population mean difference hours sleep = 0 H 1 : population mean difference hours sleep 0 We can use a one-sample t-test. Ratio paired t-test: If ratio (treatment/control) seems better to quantify effect of treatment. 6/42

7 Two related groups Example SPSS output: Concentration of antibody (µg/ml) to type II group B Streptococcus in 20 volunteers before and after immunisation (Bland and Altman, 2009). Paired Samples Test Paired Differences... Mean Std. Deviation Lower Pair Paired Samples Test Upper t df Sig. (2-tailed) Pair /42

8 Two related groups Wilcoxon signed ranks test (nonparametric): H 0 : population median difference between pairs = parameter value For the hours of sleep example: H 0 : population median difference hours sleep = 0 H 1 : population median difference hours sleep 0 Intuition: if the population median difference is zero, then approximately half of the values of differences between the two samples should be below zero. No normality required, but it does assume ( rough) symmetry. Robust to outliers Applicable to numerical and ordinal data 8/42

9 Two related groups Wilcoxon signed ranks test (nonparametric): How does it work? Compute the difference between pairs and ignore differences equal to the parameter value (sample size is reduced to n r ) Assign ranks to the absolute values of the differences (for ties, calculate the average rank) Reassign the + and - signs to the ranks Sum up positive ranks and negative ranks separately Test statistic W is the smaller value of the sum of positive ranks and sum of negative ranks Reject H0 if: Small sample size: W c with c > 0 Large sample size: Z < c or Z > c where Z = W µ W nr (nr +1), µ σ W =, W 4 nr (n σ W = r +1)(2n r +1) ; 24 W is approximately normally distributed. 9/42

10 Two related groups Wilcoxon signed ranks test (nonparametric): H0 : population median difference between pairs = 0 group 1 group 2 diff abs(diff) rank sign neg pos neg pos sum of positive ranks = 7; sum of negative ranks = 3 10/42

11 Two related groups Example SPSS output: Response of serum antigen level to AZT in 20 AIDS patients (Makutch and Parks, 1988) Test Statistics a Z Asymp. Sig. (2-tailed) Exact Sig. (2-tailed) Exact Sig. (1-tailed) Point Probability b a. b. 11/42

12 Two related groups The sign test (nonparametric): H 0 : population median difference between pairs = parameter value For the hours of sleep example: H 0 : population median difference hours sleep = 0 H 1 : population median difference hours sleep 0 No normality required, no symmetry required Robust to outliers Applicable to numerical and ordinal data Less powerful than Wilcoxon signed ranks test when the population is symmetric 12/42

13 Two related groups The sign test (nonparametric): How does it work? Compute the differences between pairs and omit differences equal to the parameter value (sample size is reduced to n r ) Count the number of positive and negative differences The test statistic W is the number of positive differences or the number of negative differences, whichever is smaller Reject H0 if: p value<α and p value=p(w or less differences) for one-sided test or p value=2p(w or less differences) for two-sided test; W has a binomial distribution with n r trials and p = 1/2. 13/42

14 Two related groups Example SPSS output: Response of serum antigen level to AZT in 20 AIDS patients (Makutch and Parks, 1988). Test Statistics a Exact Sig. (2-tailed) Exact Sig. (1-tailed) Point Probability.004 b a. b. 14/42

15 Two unrelated groups Samples from two independent (unrelated) groups of individuals. Example 1: birth weights of children born to n1 = 40 heavy smokers (group 1) and to n 2 = 42 non-smokers (group 2). Example 2: weights of two groups of children, each child being randomly allocated to receive either a dietary supplement (group 1) or placebo (group 2). 15/42

16 Two unrelated groups Unpaired (two-sample) t-test (parametric): H 0 : two populations have the same means For the birth weight example: H 0 : population mean weight group 1 = population mean weight group 2 H 1 : population mean weight group 1 population mean weight group 2 Reject H 0 if: or sample mean group 1 - sample mean group 2 SEM < c with c > 0. sample mean group 1 - sample mean group 2 SEM > c 16/42

17 Two unrelated groups Variable (e.g., weight) is numerical in each group, is normally distributed in each group and the (usually unknown) variances are the same. If population variances are equal but unknown: test statistic follows a Student-t distribution: 5% significance level: SEM = s 1/n1 + 1/n 2 and (n c = t n1 +n 2 2,1 α/2 = t n1 +n 2 2,0.975 (s = 1 1)s 1 2+(n 2 1)s 2 2 ) pooled n 1 +n 2 2 standard deviation of the two groups; s i is the standard deviations sample observations group i; n i is the sample size of group i, with i = 1, 2) 17/42

18 Two unrelated groups If population variances known (σ1 2, σ2 2 ) or sample size very large: test statistic follows a normal distribution (z-test): 5% significance level: SEM = σ 2 1 /n 1 + σ 2 2 /n 2 and c = z 1 α/2 = z = 1.96 If population variances unequal and unknown: test statistic follows a Student-t distribution: 5% significance level: SEM = s 2 1 /n 1 + s 2 2 /n 2 and c = t df,1 α/2 with df = (s2 1 /n 1+s 2 2 /n 2) 2 (s 2 1 /n 1 )2 n (s2 2 /n 2 )2 n /42

19 Two unrelated groups Example SPSS output: Galactose binding measurements for patients with Crohn disease and controls. Independent Samples Test Galactose binding Equal variances assumed Equal variances not assumed Levene's Test t-test for Equality of Means for Equality of Variances F Sig. t df Sig. (2- tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper /42

20 Two unrelated groups Wilcoxon rank sum (two-sample) test (nonparametric): H 0 : two populations have the same medians For the birth weight example: H 0 : population median weight group 1 = population median weight group 2 H 1 : population median weight group 1 population median weight group 2 No normality required but the population distribution of the two groups assumed to have the same shape Applicable to numerical and ordinal data 20/42

21 Two unrelated groups Wilcoxon rank sum (two-sample) test (nonparametric): How does it work? Rank the data ignoring grouping Sum up ranks for each group separately The test statistic W is the smaller of the sum of ranks for group 1 and sum of ranks for group 2 Reject H0 if: Small sample size: W c with c > 0 Large sample size: Z < c or Z > c where Z = W µ W, µ σ W = n 1(n 1 +n 2 +1), W 2 n1 n σ W = 2 (n 1 +n 2 +1), n 12 1 is the sample size of the group that has smaller sum of ranks; W is approximately normally distributed. 21/42

22 Two unrelated groups Mann-Whitney U (two-sample) test (nonparametric): H 0 : two populations have the same medians For the birth weight example: H 0 : population median weight group 1 = population median weight group 2 H 1 : population median weight group 1 population median weight group 2 No normality required but the population distribution of the two groups assumed to have the same shape Applicable to numerical and ordinal data 22/42

23 Two unrelated groups Mann-Whitney U (two-sample) test (nonparametric): How does it work? Rank the data ignoring grouping Sum up ranks for each group separately (R 1 and R 2 ) The test statistic U is the smaller value of U 1 and U 2 : U 1 = n 1 n 2 + n 1(n 1 + 1) 2 R 1 U 2 = n 1 n 2 + n 2(n 2 + 1) R 2 2 where n 1, n 2 are the sample sizes of the groups. Reject H0 if: Small sample size: U c with c > 0 Large sample size: Z < c or Z > c where Z = U µ U, µ σ U = n 1n 2 U 2, n1 n σ U = 2 (n 1 +n 2 +1) ; 12 U is approximately normally distributed. 23/42

24 Two unrelated groups Wilcoxon rank sum test and Mann-Whitney U test give identical results Example SPSS output: Data on diastolic blood pressure (mm Hg) measured in 4 treated subjects and 11 controls. Test Statistics a Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed) Exact Sig. (2-tailed) Exact Sig. (1-tailed) Point Probability b a. b. 24/42

25 More than two related groups Related groups: One group of individuals and variable measured on each individual in more than two circumstances. E.g. measurement while taking low dose of a drug, high dose of a drug or placebo. More than two samples of different individuals but linked to each other. Making pairwise comparisons between groups is not efficient because the type I error rate becomes high Better to carry out a single global test to determine whether the means/medians differ in ANY groups. 25/42

26 More than two related groups Friedman s ANOVA (nonparametric): H 0 : population medians are the same across groups H 1 : at least one median is different from others Example: we measure the outcome variable x of n individuals at k different conditions or k different time points. No normality required Applicable to numerical and ordinal data 26/42

27 More than two related groups Friedman s ANOVA (nonparametric) How does it work? Rank the data separately for each individual Sum up ranks for each group separately (R j for group j) The test statistic F is F = 12 R 2 j 3n(k + 1) nk(k + 1) Reject H0 if: Small sample size: F c with c > 0 Large sample size: F < c or F > c with c > 0 knowing that F has a chi-squared distribution with df=k 1 j 27/42

28 More than two unrelated groups Examples: Samples from 3 independent groups of patients, each with a type of sickle cell disease. For each patient, the steady-state haemoglobin levels are measured. RNA samples from 12 mice of 3 different strains (4 mice/strain). Identify genes that differ in expression levels among these strains. Making pairwise comparisons between groups (e.g. with t-test) is not efficient because the type I error rate becomes high Better to carry out a single global test to determine whether the means/medians differ in ANY groups. 28/42

29 More than two unrelated groups One-way analysis of variance (ANOVA) (parametric): H 0 : all populations/groups have the same means H 1 : at least one population/group has different mean than others (Homogeneity or heterogeneity across populations/groups) Example: the populations means haemoglobin level for each type of sickle cell disease are the same, or at least one is different. Variable (e.g., haemoglobin level, gene expression) is numerical is normally distributed in each group and variances are the same across groups moderate departures from normality may be ignored but unequal variances cannot check homogeneity of variances. Groups are defined by the levels of a single factor (e.g. different sickle cell disease; gender). 29/42

30 More than two unrelated groups One-way analysis of variance (ANOVA) (parametric): We measure the outcome variable x (e.g. haemoglobin level) and compare its mean in the k groups defined by the levels of a single factor (e.g. type of sickle cell disease). The outcome is measured n times in total. The variance of all observations ignoring subdivision into groups (total sample variance) is s 2 = j i (x ij x) 2 /(n 1) One-way ANOVA partitions the sum of squares SS = j i (x ij x) 2 (n - 1 degrees of freedom) into: Between-groups SS (k - 1 d.f.): SS M = j n j ( x j x) 2 Within-groups SS or residual SS (n - k d.f.): SS R = j i (x ij x j ) 2 x ij is the i observation in j group, x j is the mean of group j, x is the grand mean The amount of variation per degree of freedom is the mean square (MS). 30/42

31 More than two unrelated groups One-way analysis of variance (ANOVA) (parametric): Reject H0 at the 5% significance level if: F = Between-groups SS/(k 1) Within-groups SS/(n 1) = Between-groups MS Within-groups MS > F k 1,n k,0.95 Intuitively: if observed differences in mean haemoglobin levels for the different types of sickle cell disease were simply due to chance Between-group MS Within-group MS When there are only 2 groups, results of the one-way ANOVA exactly equal to results t-test. 31/42

32 More than two unrelated groups Example SPSS output: Galactose binding measurements for patients with Crohn disease, ulcerative colitis and controls. Galactose binding ANOVA Between Groups Within Groups Total df Mean Square F Sig /42

33 More than two unrelated groups Kruskal-Wallis test (nonparametric): H 0 : all populations/groups have the same medians H 1 : at least one population/group has different median than others The population distribution of all groups assumed to have the same shape We measure the outcome variable x (e.g. haemoglobin level) and compare its median in the k groups defined by the levels of a single factor (e.g. type of sickle cell disease). The outcome is measured n times in total. No normality required Applicable to numerical or ordinal data 33/42

34 More than two unrelated groups Kruskal-Wallis test (nonparametric): How does it work? Rank the data ignoring grouping Sum up ranks for each group separately (R j ) The test statistic H is H = 12 Rj 2 3(n + 1) n(n + 1) Reject H0 if: Small sample size: H c with c > 0 Larger sample size: H < c or H > c with c > 0 knowing that H has a chi-squared distribution with df=k 1 j n j 34/42

35 Multiple comparisons Multiple comparisons: compare all different pairwise combinations of the groups With k groups we have k(k 1) 2 pairs of groups to compare There are methods that control for the increased familywise error rate, i.e., make all pairwise comparisons while maintaining the experimentwise error rate at the pre-established α level 35/42

36 Multiple comparisons SPSS Post Hoc Multiple Comparisons option 36/42

37 Multiple comparisons Fisher s least significant difference (LSD) test: does not correct for multiple comparisons, is equivalent to performing multiple t-tests on the data Bonferroni method: for each pairwise comparison α/m is used as a significance level and overall Type I error rate is α; m is the number of all possible comparisons in SPSS: each p-value for Bonferroni test is a p-value for LSD test multiplied by the number of comparisons powerful method for a small number of comparisons 37/42

38 Multiple comparisons Tukey s HSD(Honestly Significance Difference) test: for each pairwise comparison the test statistic Q = x i x k is used where MSw /n i k, x i and x k are the group means we compare, n is the sample size of each group and MS w is the within-groups variance value from, e.g, the ANOVA method we obtained at the first phase correction for unequal sized groups: MSw is divided by m n h = 1/n 1 +1/n /n m powerful method for a large number of comparisons Dunnett s test: makes pairwise comparisons of each group to a control or reference group so we have k 1 comparisons 38/42

39 Multiple comparisons Example SPSS output: Galactose binding measurements for patients with Crohn disease, ulcerative colitis and controls. Dependent Variable: Galactose binding Multiple Comparisons (I) group (J) group Std. Error Sig. Bonferroni Crohn disease Ulcerative colitis Control * Ulcerative colitis Crohn disease Control Control Crohn disease * Ulcerative colitis Dunnett t (2-sided) b Crohn disease Control * Ulcerative colitis Control Multiple Comparisons Dependent Variable: Galactose binding 95% Confidence Interval 39/42

40 Summary of tests 2 related groups (e.g. X treatment, X placebo measured for each patient) Xtreatment X placebo numerical and normally distributed: one-sample t-test (on mean of difference) [Parametric] Xtreatment X placebo numerical or ordinal (+roughly symmetric for Wilcoxon signed-rank): Wilcoxon signed-rank test, signed test (on median of difference) [Nonparametric] Use exact version of test if n independent groups (e.g. X measured for men and women) X numerical and normally distributed for each group: independent samples t-test for homogeneous (or not) variances (on means) [Parametric] X numerical or ordinal: Mann-Whitney or Wilcoxon rank sum test (on medians if assuming distribution of X in all groups has same shape; otherwise on mean ranks) [Nonparametric] Use exact version of test if n /42

41 Summary of tests More than 2 related groups (e.g. X measured at three different time points) X numerical or ordinal: Friedman s ANOVA (on medians, assuming distribution of X in all groups has same shape) [ Nonparametric] More than 2 independent groups (e.g. X measured for three age groups) X numerical and normally distributed for each group, homogeneous variances: one-way ANOVA (on means) [Parametric] X numerical or ordinal: Kruskal-Wallis (on medians, assuming distribution of X in all groups has same shape) [ Nonparametric] Use exact version of test if n 15 and number of groups 4. 41/42

42 Parametric or nonparametric? Check the following (if groups are dependent, check it for the difference, e.g. X treatment X placebo for each patient; if groups are independent, check it for each of them separately, e.g. X in men and X in women): n < 15 (small sample): if population is known to be normally distributed (past studies, etc.) and sample histogram looks roughly normal. 15 n < 30 (moderate sample): if population is known to be symmetric and sample histogram looks symmetric too n 30 (large sample): no further checks needed, unless population and/or sample histogram is very badly skewed If for X treatment X placebo (dependent groups)/x in each group (independent groups) these checks do not lead to problems, then use corresponding t-test. If not, use nonparametric option (power might be low, though, for small samples). Check outliers. 42/42

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