Statistical Inference for more than two groups Peter T. Donnan Professor of Epidemiology and Biostatistics Statistics for Health Statistics for Health Research Research
Jan 04, 2016
Statistical Inference for more than two groups Peter T. DonnanProfessor of Epidemiology and BiostatisticsStatistics for Health Research
Tests to be coveredChi-squared testOne-way ANOVALogrank test
Significance testing general overview
Define the null and alternative hypotheses under the study
Acquire data
Calculate the value of the test statistic
Compare the value of the test statistic to values from a known probability distribution
Interpret the p-value and draw conclusion
Categorical data > 2 groups
Unordered categories Nominal- Chi-squared test for association
Ordered categories - Ordinal - Chi squared test for trend
Example
Does the proportion of mothers developingpre-eclampsia vary by parity (birth order)?
Contingency table (r x c)
Pre-eclampsiaBirth Order 1st 2nd 3rd 4th No Yes1170 (79.4%)278 (84.8%)83 (86.5%)86 (92.4%) 304 (20.6%) 50 (15.2%)13 (13.5%) 7 (7.5%)
Null hypothesis: No association between pre-eclampsia and birth orderNull hypothesis: There is no trend in pre-eclampsia with parityNull Hypotheses
Test of associationTest of linear trend
Strong association between pre-eclampsia and birth order (2 = 15.42, p = 0.001)Significant linear trend in incidence of pre-eclampsia with parity (2 = 15.03, p < 0.001)Note 3 degrees of freedom for association test and 1 df for test for trendConclusions
Contingency table (r x c)
Pre-eclampsiaBirth Order 1st 2nd 3rd 4th No Yes1170 (79.4%)278 (84.8%)83 (86.5%)86 (92.4%) 304 (20.6%) 50 (15.2%)13 (13.5%) 7 (7.5%)
Tables can be any size. For example SIMD deciles by parity would be a 10 x 4 tableBut with very large tables difficult to interpret tests of associationCrosstabulations in SPSS can give Odds ratios as an option with row or column with two categoriesContingency Tables (r x c)
Numerical data > 2 groups
Compare means from several groups
Single global test of difference in meansAlso test for linear trend
1-way analysis of variance (ANOVA)
Extend t-test to >2 groups i.e Analysis of Variance (ANOVA)Consider scores for contribution to energy intake from fat groups, milk groups and alcohol groups
Does the mean score differ across the three categories of intake groups?Koh ET, Owen WL. Introduction to Nutrition and Health Research Kluwer Boston, 2000
One-Way ANOVA of scoresContributor to Energy Intake Alcoholn=6Mean=4.22n=6Mean=0.167FatMilkn=6Mean=2.01
One-Way ANOVA of ScoresThe null hypothesis (H0) is there are no differences in mean score across the three groupsUse SPSS One-Way ANOVA to carry out this test
Assumptions of 1-Way ANOVA1. Standard deviations are similar2. Test variable (scores) are approx. Normally distributedIf assumptions are not met, use non-parametric equivalent Kruskal-Wallis test
Results of ANOVA ANOVA partitions variation into Within and Between group components
Results in F-statistic compared with values in F-tables
F = 108.6, with 2 and 15 df, p
Results of ANOVA The groups differ significantly and it is clear the Fat group contributes most to energy score with a mean = 4.22
Further pair-wise comparisons can be made (3 possible) using multiple comparisons test e.g. Bonferroni
Example 2
Does income vary by highest levelof education achieved?
H0: no difference in mean income by education level achieved
H1: mean income varies with education level achievedNull Hypothesis and alternative
Assumptions of 1-Way ANOVAStandard deviations or variances are similar Test variable (income) are approx. Normally distributed
If assumptions are not met, use non-parametric equivalent Kruskal-Wallis test
Table of Mean income for each level of educational achievement
Analysis of Variance TableF-test givesP < 0.001 showing significant difference between mean levels of education
Table of each pairwise comparison.Note lower income for did not complete school to all other groups.All p-values adjusted for multiple comparisons
Summary of ANOVA ANOVA useful if number of groups with continuous summary in eachSPSS does all pairwise group comparisons adjusted for multiple testingNote that ANOVA is just a form of linear regression see later
Extending Kaplan-Meier and logrank test in SPSSYou need to specify:Survival time time from surgery (tfsurg)Status Dead = 1, censored = 0 (dead)Factor Dukes stage at baseline (A, B, C, D, Unknown)Select compare factor and logrank Optionally select plot of survival
Implementing Logrank test in SPSS
Select options to obtain plot and median survivalSelect Compare Factor to obtain logrank testSelect linear trend for this test
Overall Comparisons Chi-Square dfSig.Log Rank (Mantel-Cox) 80.534 1.000The vector of trend weights is -2, -1, 0, 1, 2. This is the default.The test for trend in survival across Dukes stage is highly significant
Interpret SPSS outputNote the logrank statistic, degrees of freedom and statistical significance (p-value).Note in which direction survival is worst or best and back up visual information from the Kaplan-Meier plot with median survival and 95% confidence intervals from the output.Finally, interpret the results!
Interpret test result in relation to median survival
Dukes StageMedian Survival (days)Mean Survival(Days)A27701978B17491866C11201304D375646Unknown5811297
Output form Kaplan-Meier in SPSSNote that SPSS gives three possible tests:Logrank, Tarone-Ware and BreslowIn general, logrank gives greater weight to later events compared to the other two tests. If all are similar quote logrank test.If different results, quote more than one test result
Editing SPSS outputNote that everything in the SPSS output window can be copied and pasted into Word and Powerpoint.Double-clicking on plots also allows editing of the plot such as changing axes, colours, fonts, etc.
Diabetic patients LDL dataTry carrying out extended Crosstabulations and ANOVA where appropriate in the LDL dataE.g. APOE genotype
Colorectal cancer patients: survival following surgery Try carrying out Kaplan-Meier plots and logrank tests for other factors such as WHO Functional Performance, smoking, etc
Extending test to more than 2 groups Summary
Define H0 and H1
Choosing the appropriate test according to type of variables
Interpret output carefully