# Analysis of numerical data S4

Save this PDF as:

Size: px
Start display at page:

## Transcription

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

### c. The factor is the type of TV program that was watched. The treatment is the embedded commercials in the TV programs.

STAT E-150 - Statistical Methods Assignment 9 Solutions Exercises 12.8, 12.13, 12.75 For each test: Include appropriate graphs to see that the conditions are met. Use Tukey's Honestly Significant Difference

### INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of

### SPSS Tests for Versions 9 to 13

SPSS Tests for Versions 9 to 13 Chapter 2 Descriptive Statistic (including median) Choose Analyze Descriptive statistics Frequencies... Click on variable(s) then press to move to into Variable(s): list

### Statistiek I. t-tests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35

Statistiek I t-tests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistiek-i/ John Nerbonne 1/35 t-tests To test an average or pair of averages when σ is known, we

### ANOVA Analysis of Variance

ANOVA Analysis of Variance What is ANOVA and why do we use it? Can test hypotheses about mean differences between more than 2 samples. Can also make inferences about the effects of several different IVs,

### Testing Hypotheses using SPSS

Is the mean hourly rate of male workers \$2.00? T-Test One-Sample Statistics Std. Error N Mean Std. Deviation Mean 2997 2.0522 6.6282.2 One-Sample Test Test Value = 2 95% Confidence Interval Mean of the

### 1.5 Oneway Analysis of Variance

Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

### How to choose a statistical test. Francisco J. Candido dos Reis DGO-FMRP University of São Paulo

How to choose a statistical test Francisco J. Candido dos Reis DGO-FMRP University of São Paulo Choosing the right test One of the most common queries in stats support is Which analysis should I use There

### 3. Nonparametric methods

3. Nonparametric methods If the probability distributions of the statistical variables are unknown or are not as required (e.g. normality assumption violated), then we may still apply nonparametric tests

### Testing: is my coin fair?

Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible

### Outline of Topics. Statistical Methods I. Types of Data. Descriptive Statistics

Statistical Methods I Tamekia L. Jones, Ph.D. (tjones@cog.ufl.edu) Research Assistant Professor Children s Oncology Group Statistics & Data Center Department of Biostatistics Colleges of Medicine and Public

### Lecture 7: Binomial Test, Chisquare

Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two

### Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction

### Rank-Based Non-Parametric Tests

Rank-Based Non-Parametric Tests Reminder: Student Instructional Rating Surveys You have until May 8 th to fill out the student instructional rating surveys at https://sakai.rutgers.edu/portal/site/sirs

### UNDERSTANDING THE ONE-WAY ANOVA

UNDERSTANDING The One-way Analysis of Variance (ANOVA) is a procedure for testing the hypothesis that K population means are equal, where K >. The One-way ANOVA compares the means of the samples or groups

### Statistics. One-two sided test, Parametric and non-parametric test statistics: one group, two groups, and more than two groups samples

Statistics One-two sided test, Parametric and non-parametric test statistics: one group, two groups, and more than two groups samples February 3, 00 Jobayer Hossain, Ph.D. & Tim Bunnell, Ph.D. Nemours

### THE KRUSKAL WALLLIS TEST

THE KRUSKAL WALLLIS TEST TEODORA H. MEHOTCHEVA Wednesday, 23 rd April 08 THE KRUSKAL-WALLIS TEST: The non-parametric alternative to ANOVA: testing for difference between several independent groups 2 NON

### An example ANOVA situation. 1-Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters)

An example ANOVA situation Example (Treating Blisters) 1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College Subjects: 25 patients with blisters Treatments: Treatment A, Treatment

### Hypothesis testing S2

Basic medical statistics for clinical and experimental research Hypothesis testing S2 Katarzyna Jóźwiak k.jozwiak@nki.nl 2nd November 2015 1/43 Introduction Point estimation: use a sample statistic to

### Hypothesis Testing. Dr. Bob Gee Dean Scott Bonney Professor William G. Journigan American Meridian University

Hypothesis Testing Dr. Bob Gee Dean Scott Bonney Professor William G. Journigan American Meridian University 1 AMU / Bon-Tech, LLC, Journi-Tech Corporation Copyright 2015 Learning Objectives Upon successful

### Non-Parametric Two-Sample Analysis: The Mann-Whitney U Test

Non-Parametric Two-Sample Analysis: The Mann-Whitney U Test When samples do not meet the assumption of normality parametric tests should not be used. To overcome this problem, non-parametric tests can

### 6 Comparison of differences between 2 groups: Student s T-test, Mann-Whitney U-Test, Paired Samples T-test and Wilcoxon Test

6 Comparison of differences between 2 groups: Student s T-test, Mann-Whitney U-Test, Paired Samples T-test and Wilcoxon Test Having finally arrived at the bottom of our decision tree, we are now going

### CHAPTER 14 NONPARAMETRIC TESTS

CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences

### EPS 625 INTERMEDIATE STATISTICS FRIEDMAN TEST

EPS 625 INTERMEDIATE STATISTICS The Friedman test is an extension of the Wilcoxon test. The Wilcoxon test can be applied to repeated-measures data if participants are assessed on two occasions or conditions

### Study Guide for the Final Exam

Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

### Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

### Module 9: Nonparametric Tests. The Applied Research Center

Module 9: Nonparametric Tests The Applied Research Center Module 9 Overview } Nonparametric Tests } Parametric vs. Nonparametric Tests } Restrictions of Nonparametric Tests } One-Sample Chi-Square Test

### L.8: Analysing continuous data

L.8: Analysing continuous data - Types of variables - Comparing two means: - independent samples - Comparing two means: - dependent samples - Checking the assumptions - Nonparametric test Types of variables

### Contents 1. Contents

Contents 1 Contents 3 K-sample Methods 2 3.1 Setup............................ 2 3.2 Classic Method Based on Normality Assumption..... 3 3.3 Permutation F -test.................... 5 3.4 Kruskal-Wallis

### Research Methodology: Tools

MSc Business Administration Research Methodology: Tools Applied Data Analysis (with SPSS) Lecture 11: Nonparametric Methods May 2014 Prof. Dr. Jürg Schwarz Lic. phil. Heidi Bruderer Enzler Contents Slide

### 4.4. Further Analysis within ANOVA

4.4. Further Analysis within ANOVA 1) Estimation of the effects Fixed effects model: α i = µ i µ is estimated by a i = ( x i x) if H 0 : µ 1 = µ 2 = = µ k is rejected. Random effects model: If H 0 : σa

### Chapter 16 Appendix. Nonparametric Tests with Excel, JMP, Minitab, SPSS, CrunchIt!, R, and TI-83-/84 Calculators

The Wilcoxon Rank Sum Test Chapter 16 Appendix Nonparametric Tests with Excel, JMP, Minitab, SPSS, CrunchIt!, R, and TI-83-/84 Calculators These nonparametric tests make no assumption about Normality.

### 13: Additional ANOVA Topics. Post hoc Comparisons

13: Additional ANOVA Topics Post hoc Comparisons ANOVA Assumptions Assessing Group Variances When Distributional Assumptions are Severely Violated Kruskal-Wallis Test Post hoc Comparisons In the prior

### 1 Nonparametric Statistics

1 Nonparametric Statistics When finding confidence intervals or conducting tests so far, we always described the population with a model, which includes a set of parameters. Then we could make decisions

### Inferences About Differences Between Means Edpsy 580

Inferences About Differences Between Means Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Inferences About Differences Between Means Slide

### SPSS Workbook 4 T-tests

TEESSIDE UNIVERSITY SCHOOL OF HEALTH & SOCIAL CARE SPSS Workbook 4 T-tests Research, Audit and data RMH 2023-N Module Leader:Sylvia Storey Phone:016420384969 s.storey@tees.ac.uk SPSS Workbook 4 Differences

### Testing for differences I exercises with SPSS

Testing for differences I exercises with SPSS Introduction The exercises presented here are all about the t-test and its non-parametric equivalents in their various forms. In SPSS, all these tests can

### UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally

### Biostatistics. ANOVA - Analysis of Variance. Burkhardt Seifert & Alois Tschopp. Biostatistics Unit University of Zurich

Biostatistics ANOVA - Analysis of Variance Burkhardt Seifert & Alois Tschopp Biostatistics Unit University of Zurich Master of Science in Medical Biology 1 ANOVA = Analysis of variance Analysis of variance

### Statistics and research

Statistics and research Usaneya Perngparn Chitlada Areesantichai Drug Dependence Research Center (WHOCC for Research and Training in Drug Dependence) College of Public Health Sciences Chulolongkorn University,

### Week 7 Lecture: Two-way Analysis of Variance (Chapter 12) Two-way ANOVA with Equal Replication (see Zar s section 12.1)

Week 7 Lecture: Two-way Analysis of Variance (Chapter ) We can extend the idea of a one-way ANOVA, which tests the effects of one factor on a response variable, to a two-way ANOVA which tests the effects

### ANSWERS TO EXERCISES AND REVIEW QUESTIONS

ANSWERS TO EXERCISES AND REVIEW QUESTIONS PART FIVE: STATISTICAL TECHNIQUES TO COMPARE GROUPS Before attempting these questions read through the introduction to Part Five and Chapters 16-21 of the SPSS

### Nonparametric Test Procedures

Nonparametric Test Procedures 1 Introduction to Nonparametrics Nonparametric tests do not require that samples come from populations with normal distributions or any other specific distribution. Hence

### Parametric and non-parametric statistical methods for the life sciences - Session I

Why nonparametric methods What test to use? Rank Tests Parametric and non-parametric statistical methods for the life sciences - Session I Liesbeth Bruckers Geert Molenberghs Interuniversity Institute

### The Statistics Tutor s

statstutor community project encouraging academics to share statistics support resources All stcp resources are released under a Creative Commons licence Stcp-marshallowen-7 The Statistics Tutor s www.statstutor.ac.uk

### Introduction to Analysis of Variance (ANOVA) Limitations of the t-test

Introduction to Analysis of Variance (ANOVA) The Structural Model, The Summary Table, and the One- Way ANOVA Limitations of the t-test Although the t-test is commonly used, it has limitations Can only

### Nonparametric Statistics

Nonparametric Statistics J. Lozano University of Goettingen Department of Genetic Epidemiology Interdisciplinary PhD Program in Applied Statistics & Empirical Methods Graduate Seminar in Applied Statistics

### SPSS Guide: Tests of Differences

SPSS Guide: Tests of Differences I put this together to give you a step-by-step guide for replicating what we did in the computer lab. It should help you run the tests we covered. The best way to get familiar

### Chapter 7. One-way ANOVA

Chapter 7 One-way ANOVA One-way ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The t-test of Chapter 6 looks

### SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES

SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR

### Chapter 3: Nonparametric Tests

B. Weaver (15-Feb-00) Nonparametric Tests... 1 Chapter 3: Nonparametric Tests 3.1 Introduction Nonparametric, or distribution free tests are so-called because the assumptions underlying their use are fewer

### MEASURES OF LOCATION AND SPREAD

Paper TU04 An Overview of Non-parametric Tests in SAS : When, Why, and How Paul A. Pappas and Venita DePuy Durham, North Carolina, USA ABSTRACT Most commonly used statistical procedures are based on the

### Non-parametric tests I

Non-parametric tests I Objectives Mann-Whitney Wilcoxon Signed Rank Relation of Parametric to Non-parametric tests 1 the problem Our testing procedures thus far have relied on assumptions of independence,

### Nonparametric Methods for Two Samples. Nonparametric Methods for Two Samples

Nonparametric Methods for Two Samples An overview In the independent two-sample t-test, we assume normality, independence, and equal variances. This t-test is robust against nonnormality, but is sensitive

### Comparing three or more groups (one-way ANOVA...)

Page 1 of 36 Comparing three or more groups (one-way ANOVA...) You've measured a variable in three or more groups, and the means (and medians) are distinct. Is that due to chance? Or does it tell you the

### Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures

Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone:

### Non-parametric Tests and some data from aphasic speakers

Non-parametric Tests and some data from aphasic speakers Vasiliki Koukoulioti Seminar Methodology and Statistics 19th March 2008 Some facts about non-parametric tests When to use non-parametric tests?

### Nonparametric Statistics

1 14.1 Using the Binomial Table Nonparametric Statistics In this chapter, we will survey several methods of inference from Nonparametric Statistics. These methods will introduce us to several new tables

### Nonparametric tests these test hypotheses that are not statements about population parameters (e.g.,

CHAPTER 13 Nonparametric and Distribution-Free Statistics Nonparametric tests these test hypotheses that are not statements about population parameters (e.g., 2 tests for goodness of fit and independence).

### Analysis of Data. Organizing Data Files in SPSS. Descriptive Statistics

Analysis of Data Claudia J. Stanny PSY 67 Research Design Organizing Data Files in SPSS All data for one subject entered on the same line Identification data Between-subjects manipulations: variable to

### Stat 371, Cecile Ane Practice problems Midterm #2, Spring 2012

Stat 371, Cecile Ane Practice problems Midterm #2, Spring 2012 The first 3 problems are taken from previous semesters exams, with solutions at the end of this document. The other problems are suggested

### Nonparametric Two-Sample Tests. Nonparametric Tests. Sign Test

Nonparametric Two-Sample Tests Sign test Mann-Whitney U-test (a.k.a. Wilcoxon two-sample test) Kolmogorov-Smirnov Test Wilcoxon Signed-Rank Test Tukey-Duckworth Test 1 Nonparametric Tests Recall, nonparametric

### Nonparametric tests, Bootstrapping

Nonparametric tests, Bootstrapping http://www.isrec.isb-sib.ch/~darlene/embnet/ Hypothesis testing review 2 competing theories regarding a population parameter: NULL hypothesis H ( straw man ) ALTERNATIVEhypothesis

### INTERPRETING THE REPEATED-MEASURES ANOVA

INTERPRETING THE REPEATED-MEASURES ANOVA USING THE SPSS GENERAL LINEAR MODEL PROGRAM RM ANOVA In this scenario (based on a RM ANOVA example from Leech, Barrett, and Morgan, 2005) each of 12 participants

### ANOVA - Analysis of Variance

ANOVA - Analysis of Variance ANOVA - Analysis of Variance Extends independent-samples t test Compares the means of groups of independent observations Don t be fooled by the name. ANOVA does not compare

### Using SPSS version 14 Joel Elliott, Jennifer Burnaford, Stacey Weiss

Using SPSS version 14 Joel Elliott, Jennifer Burnaford, Stacey Weiss SPSS is a program that is very easy to learn and is also very powerful. This manual is designed to introduce you to the program however,

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### Chi-Square P216; 269

Chi-Square P16; 69 Confidence intervals CI: % confident that interval contains population mean (µ) % is determined by researcher (e.g. 85, 90, 95%) Formula for z-test and t-test: CI = M +/- z*(σ M ) CI

### One-Way Analysis of Variance

Spring, 000 - - Administrative Items One-Way Analysis of Variance Midterm Grades. Make-up exams, in general. Getting help See me today -:0 or Wednesday from -:0. Send an e-mail to stine@wharton. Visit

### Statistics: revision

NST 1B Experimental Psychology Statistics practical 5 Statistics: revision Rudolf Cardinal & Mike Aitken 3 / 4 May 2005 Department of Experimental Psychology University of Cambridge Slides at pobox.com/~rudolf/psychology

### SPSS 3: COMPARING MEANS

SPSS 3: COMPARING MEANS UNIVERSITY OF GUELPH LUCIA COSTANZO lcostanz@uoguelph.ca REVISED SEPTEMBER 2012 CONTENTS SPSS availability... 2 Goals of the workshop... 2 Data for SPSS Sessions... 3 Statistical

### Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 2000: Page 1:

Research Methods 1 Handouts, Graham Hole,COGS - version 1.0, September 000: Page 1: NON-PARAMETRIC TESTS: What are non-parametric tests? Statistical tests fall into two kinds: parametric tests assume that

### Wilcoxon Rank Sum or Mann-Whitney Test Chapter 7.11

STAT Non-Parametric tests /0/0 Here s a summary of the tests we will look at: Setting Normal test NonParametric Test One sample One-sample t-test Sign Test Wilcoxon signed-rank test Matched pairs Apply

### One Way ANOVA. A method for comparing several means along a single variable

Analysis of Variance (ANOVA) One Way ANOVA A method for comparing several means along a single variable It is the same as an independent samples t test, test but for 3 or more samples Called one way when

### Overview of Non-Parametric Statistics PRESENTER: ELAINE EISENBEISZ OWNER AND PRINCIPAL, OMEGA STATISTICS

Overview of Non-Parametric Statistics PRESENTER: ELAINE EISENBEISZ OWNER AND PRINCIPAL, OMEGA STATISTICS About Omega Statistics Private practice consultancy based in Southern California, Medical and Clinical

### We know from STAT.1030 that the relevant test statistic for equality of proportions is:

2. Chi 2 -tests for equality of proportions Introduction: Two Samples Consider comparing the sample proportions p 1 and p 2 in independent random samples of size n 1 and n 2 out of two populations which

### The bread and butter of statistical data analysis is the Student s t-test.

03-Elliott-4987.qxd 7/18/2006 3:43 PM Page 47 3 Comparing One or Two Means Using the t-test The bread and butter of statistical data analysis is the Student s t-test. It was named after a statistician

### 1/22/2016. What are paired data? Tests of Differences: two related samples. What are paired data? Paired Example. Paired Data.

Tests of Differences: two related samples What are paired data? Frequently data from ecological work take the form of paired (matched, related) samples Before and after samples at a specific site (or individual)

### Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

### Independent t- Test (Comparing Two Means)

Independent t- Test (Comparing Two Means) The objectives of this lesson are to learn: the definition/purpose of independent t-test when to use the independent t-test the use of SPSS to complete an independent

### NONPARAMETRIC STATISTICS 1. depend on assumptions about the underlying distribution of the data (or on the Central Limit Theorem)

NONPARAMETRIC STATISTICS 1 PREVIOUSLY parametric statistics in estimation and hypothesis testing... construction of confidence intervals computing of p-values classical significance testing depend on assumptions

### Math 62 Statistics Sample Exam Questions

Math 62 Statistics Sample Exam Questions 1. (10) Explain the difference between the distribution of a population and the sampling distribution of a statistic, such as the mean, of a sample randomly selected

### On Importance of Normality Assumption in Using a T-Test: One Sample and Two Sample Cases

On Importance of Normality Assumption in Using a T-Test: One Sample and Two Sample Cases Srilakshminarayana Gali, SDM Institute for Management Development, Mysore, India. E-mail: lakshminarayana@sdmimd.ac.in

### Chapter 2 Probability Topics SPSS T tests

Chapter 2 Probability Topics SPSS T tests Data file used: gss.sav In the lecture about chapter 2, only the One-Sample T test has been explained. In this handout, we also give the SPSS methods to perform

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Some Critical Information about SOME Statistical Tests and Measures of Correlation/Association

Some Critical Information about SOME Statistical Tests and Measures of Correlation/Association This information is adapted from and draws heavily on: Sheskin, David J. 2000. Handbook of Parametric and

### Tutorial 5: Hypothesis Testing

Tutorial 5: Hypothesis Testing Rob Nicholls nicholls@mrc-lmb.cam.ac.uk MRC LMB Statistics Course 2014 Contents 1 Introduction................................ 1 2 Testing distributional assumptions....................

### Variables and Data A variable contains data about anything we measure. For example; age or gender of the participants or their score on a test.

The Analysis of Research Data The design of any project will determine what sort of statistical tests you should perform on your data and how successful the data analysis will be. For example if you decide

### Reporting Statistics in Psychology

This document contains general guidelines for the reporting of statistics in psychology research. The details of statistical reporting vary slightly among different areas of science and also among different

### Statistiek II. John Nerbonne. October 1, 2010. Dept of Information Science j.nerbonne@rug.nl

Dept of Information Science j.nerbonne@rug.nl October 1, 2010 Course outline 1 One-way ANOVA. 2 Factorial ANOVA. 3 Repeated measures ANOVA. 4 Correlation and regression. 5 Multiple regression. 6 Logistic

### Chapter 7 Section 7.1: Inference for the Mean of a Population

Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used

### Chapter 21 Section D

Chapter 21 Section D Statistical Tests for Ordinal Data The rank-sum test. You can perform the rank-sum test in SPSS by selecting 2 Independent Samples from the Analyze/ Nonparametric Tests menu. The first

### One-sample normal hypothesis Testing, paired t-test, two-sample normal inference, normal probability plots

1 / 27 One-sample normal hypothesis Testing, paired t-test, two-sample normal inference, normal probability plots Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### Comparing Means in Two Populations

Comparing Means in Two Populations Overview The previous section discussed hypothesis testing when sampling from a single population (either a single mean or two means from the same population). Now we

### EPS 625 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM

EPS 6 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM ANCOVA One Continuous Dependent Variable (DVD Rating) Interest Rating in DVD One Categorical/Discrete Independent Variable

### HYPOTHESIS TESTING (TWO SAMPLE) - CHAPTER 8 1. how can a sample be used to estimate the unknown parameters of a population

HYPOTHESIS TESTING (TWO SAMPLE) - CHAPTER 8 1 PREVIOUSLY estimation how can a sample be used to estimate the unknown parameters of a population use confidence intervals around point estimates of central

### ANOVA MULTIPLE CHOICE QUESTIONS. In the following multiple-choice questions, select the best answer.

ANOVA MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, select the best answer. 1. Analysis of variance is a statistical method of comparing the of several populations. a. standard