Statistic

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Statistics are methods to analyse data of quantitative studies.

Contents 1 design and statistical method 2 methods for data analysis 2.1 Describing a single variable 2.2 Differences in variable between two groups 2.3 diferences in variable between more than two groups 2.4 differences between groups adjusted for other variables 2.5 paired changes within subjects 2.6 repeated changes within subjects 2.7 agreement within subjects (diagnostic accuracy) 2.8 association between two variables 2.9 predicting one variable from others 2.10 searching for patterns in the data 3 literature 4 weblinks

design and statistical method

The choice of statistical methods depends on:

• the analysis scenario which depend on the research questions and study design
• the measurement levels of the variables concerned

study design	common statistical methods
parallel RCT	between-group tests (e.g. chi-square-test, Mann-Whitney-U-test, t-test, Kruskal-Wallis-test, one-way analysis of variance, analysis of covariance)
non-randomised design	logistic or multiple regression adjusting for covariates
matched RCT	within-subject tests (e.g. McNerman-Test, sign test, Wilcoxon-Test, paired t-test)
cross-over RCT	t-tests, repeated measures analysis of variance
cluster RCT	logistic regression or multiple regression with clustering, multi-level-modelling
before-and-after-study	within-subject tests (e.g. McNerman-Test, sign test, Wilcoxon-Test, paired t-test)
cross-sectional survey	descriptive analysis, risk ratios, odds ratios, correlations, logistic or multiple regression adjusting for covariates
cohort-study	within-subject tests (e.g. McNerman-Test, sign test, Wilcoxon-Test, paired t-test) repeated measures analysis of variance between-group tests of summary measures(e.g. chi-square-test, Mann-Whitney-U-test, t-test, Kruskal-Wallis-test, one-way analysis of variance, analysis of covariance) time-to-event analysis
case-control-study	unconditional or conditional logistic regression
pilot/exploratory study	descriptive analysis (otherwise proceed with caution!)

methods for data analysis

Describing a single variable

Variable	Graphics	Statistics	Tests
Dichotomous	bar chart pie chart	%number in important category CI for % or proportion	binomial test for specific proportion
Nominal	bar chart pie chart	%number in each category	chi-square goodness-of-fit-test for specific proportions
ordinal	bar chart	%number in each category %number up to and including each category	Kolmogorov-Smirnov-Test for Poisson counts
interval	histogramm box plot dot plot	mean(sd) median (range or IQR) 5-number summary CI for mean or median skewness and kurtosis	Kolmogorov-Smirnov-Test & Shapiro-Wilk-Test for Normal one-sample t-test for specific mean
interval time-to-event	Kaplan-Meier curve	median survival time survival rates	?

Differences in variable between two groups

Variable	Graphics	Statistics	Tests
dichotomous	clustered or stacked bar chart	%number in important category within group risk ratio odds ratio CI for difference between two %s or proprtions	chi-square-test Fisher's exact Test
nominal	clustered or stacked bar chart	two-way table of numbers and %s within group	chi-square-test Freeman-Halto extension to Fisher's exact Test
ordinal	clustered or stacked bar chart	two-way table of numbers and %s within group	chi-square-test (order ignored) chi-square-test for trend Mann-Whitney-U-Test (few ties) two-sample Kolmogorov-Smirnov-Test (many ties)
interval	histogramm by group box plot bar chart of means/medians by group error bar chart of CIs for means/medians	mean(sd) or median(range or IQR) within group CI for difference between two means/medians	independent t-test (equal variances) Welch-Test (unequal variances) Mann-Whitney-U-Test (high skewness)
interval time-to-event	Kaplan-Meier curve	median survival time and survival rates by group	Long-rank-test

diferences in variable between more than two groups

Variable	Graphics	Statistics	Tests
dichotomous	clustered or stacked bar chart	%number in important category within group	chi-square-test Freeman-Halton extension to Fisher's exact test
nominal	clustered or stacked bar chart	two-way table of numbers and %s within group	chi-square-test Freeman-Halton extension to Fisher's exact test
ordinal	clustered or stacked bar chart	two-way table of numbers and %s within group	chi-square-test (order ignored) chi-square-test for trend Kruskal-Wallis-Test Jonckheere's Test
interval	histogram by group box plot bar chart of means/medians by group error bar chart of CIs for means/medians	mean(sd) or median(range or IQR) within group	one-way analysis of variance (equal variances) Brown-Forsythe Test or Welch-Test (unequal variances) Kruskal-Wallis-Test and Jonckheere's Test (high skewness) factorial analysis of variance
interval time-to-event	Kaplan-Meier curves	median survival time and survival rates by group	long-rank-test

differences between groups adjusted for other variables

The other variable may be baseline measurements or variables reflecting underlying differences between groups

outcome variable	explanatory variables	Tests
dichotomous	binary or interval	binary logistic regression conditional logistic regression (paired observations)
nominal	binary or interval	multinominal or polychotomous logistic regression
ordinal	binary or interval	ordinal logistic regression
interval	binary or interval	analysis of covariance multiple regression (OLS)

paired changes within subjects

1st measure	2nd measure	Graphics/Statistics	Tests
dichotomous	dichotomous	table of 1st measure vs 2nd measure number (%) changing category CI for difference between paired %s	McNemar's-Test
nominal	nominal	table of 1st measure vs 2nd measure number (%) changing category	McNemar's-Bowker-Test
ordinal	ordnial	table of 1st measure vs 2nd measure number (%) increasing or decreasing	Sign-Test
interval	interval	histogram of difference variable mean(sd) or median(range of IQR) of differnce variable CI for difference between paired means	paired t-test Wilcoxon matched-pairs-sign-rank-test (high skewness)

repeated changes within subjects

1st measure	2nd measure	Graphics/Statistics	Tests
dichotomous	dichotomous	table of 1st variable vs 2nd variable table of 1st variable vs 3nd variable etc...	Cochran's Q-Test
nominal	nominal	table of 1st variable vs 2nd variable table of 1st variable vs 3nd variable etc...	pairwise McNemar's-Bowker-Test
ordinal	ordnial	table of 1st variable vs 2nd variable table of 1st variable vs 3nd variable etc...	Friedman's-Test
interval	interval	CI for difference between paired means	repeated measures ANOVA Friedman's-Test Page's L-Test

agreement within subjects (diagnostic accuracy)

1st measure	2nd measure	Graphics/Statistics	Tests
dichotomous	dichotomous	table of 1st measure vs 2nd measure	percent agreement sensitivity and specificity ROC curve Cohen's Kappa
nominal	nominal	table of 1st measure vs 2nd measure	Cohen's Kappa weighted Kappa
ordinal	ordnial	table of 1st measure vs 2nd measure	weighted Kappa
interval	interval	Bland-Altman-Plot of difference v mean for each subject	Blant-Altman limits of agreement intraclass correlation

association between two variables

1st measure	2nd measure	Graphics/Statistics	Tests
dichotomous	dichotomous	two-way table of numbers and %s|	risk ratio odds ratio phi statistic tetrachronic correlation
dichotomus	nominal	two-way table of numbers and %s	phi statistic or Cramer's V-statistic
nominal	nominal	two-way table of numbers and %s	Cramer's V-statistic lambda
dichotomus	ordinal	two-way table of numbers and %s	Kendall's correlation Spearman's correlation probability of superiority PS=P(X<Y)
nominal	ordinal	two-way table of numbers and %s	pairwise PSs
ordinal	ordnial	two-way table of numbers and %s	Somer's d Kendall's correlation Spearman's correlation
dichotomous	interval	box plot scatter plot	point bi-serial correlation Cohen's d
nominal	interval	box plot scatter plot	eta-squared and omega-squared
ordinal	interval	box plot scatter plot	Kendall's correlation Spearman's correlation
interval	interval	scatter plot	Pearson's correlation (linear relationship) Kendall's correlation or Spearman's correlation (high skewness)

predicting one variable from others

dependent variable (DV)	independent variables (IVs)	Method
dichotomous	binary or interval	binary logistic regression
nominal	binary or interval	multinominal or polychotomous logistic regression discriminant analysis
ordinal	binary or interval	ordinal logistic regression
interval	binary or interval	multiple regression (OLS)
interval number of events in fixed time or rate of events	binary or interval	poisson regression negative binomial regression (if greater variance)
interval time to event	binary or interval	cox regression

searching for patterns in the data

searching for	variables	Method
groups of related variables	binary	factor analysis of tetrachoric correlations
groups of related variables	ordinal (e.g. Likkert)	factor analysis of Pearson correlations (usually) or covariances
groups of related variables	interval	factor analysis of Pearson correlations (usually) or covariances
groups of related subjects	binary	cluster analysis with binary distance measure
groups of related subjects	interval	cluster analysis with interval distance measure

literature

• Katz (2006): "Study Design and Statistical Analysis: A Practical Guide for Clinicians", Cambridge University Press, ISBN 978-0521534079

weblinks

• Statistics Guide for Research Grant Applicants, St George's Hospital Medical School, London
• Statistics of Europe
• Wikiversity: How to use the free statistic software R