Research Methods: 2M.Sc.

Physiotherapy/Podiatry/Pain

Inferential Statistics

Why ?

• Differences between samples/data sets

• Differences in means or medians of samples

• Different enough?

• Different by chance?

• Different due to treatment?

• Differences in ?

Testing the differences

• Differences between sample

• Relative to (Xi – )2 n

Differences in the sample Measure(s) of Centrality Relative to the variance of the

samples

x

x

High variance = big overlap

Medium variance = medium overlap

Low variance = small overlap

Inferential statistical tests

Put a value on this relationship; overlap versus difference

Test that value against expected norms

State probability of that degree of difference with that degree of overlap

The t-test

groups of Variance

meansin Differencet statistic =

t statistic is interpreted relativeto the DF for sample(s)

The t-test

21

21

-

-

xxSE

xxt statistic =

(Standard Error of the Difference)

The t-test

)1(var

)1(var

-

2

2

1

1

21

nn

xxt

The t-test

• Look up t statistic in tables of the t distribution

• Is t significant = is the difference between the two data sets significant ?

• One or two tailed test?

Two tailed: 0 or 1 2

One tailed: or 0 or 1 or 2

95%

Assumptions; t-tests

t statistic is only representative of the level of difference if data is Parametric

Interval or Ratio and Normally distributed

Only compares two samples, three or more…?

Assumptions; 1 way ANOVA

Three or more samples

One-way Analysis of Variance = One-Way ANOVA

Parametric Data which is Homoscedastic;

SPSS; Levenes test for Homogeniety of Variance

Heteroscedastic

Homoscedastic

Non-Parametric tests

• Test differences in medians or rank order

• Non Parametric equivalents of t-tests;

Mann-Whitney U-test or Wilcoxon

• Non Parametric equivalent of the One-way ANOVA;

Kruskal Wallis Test or Friedmans

Parametric or Non-Parametric ?

• Parametric = Interval or Ratio Normally Distributed

• Non-Parametric = Interval or Ratio not Normally Distributed and Nominal and Ordinal data

• So…….. Test for normality?

Test of Normality of Distribution• Normal Probability Plots; Shapiro-Wilk,

Anderson Darling, Kolmogorov Smirnov, n-Score etc

• Calculate a test statistic • SPSS: n < 50 Shapiro-Wilk; n > 50 Kolmogorov

Smirnovp > 0.05 normal p < 0.05 not normal

p values and types of errors

• Difference is significant if less than 5% probability it occurred by chance

p < 0.05

p values and types of errors

Type I (Alpha) error; There is no significant difference but you think there is.

Protection by setting high “Alpha exclusion value”

p < 0.05

p values and types of errors

Type II (Beta) error

There is a significant difference and you miss it; Study has a low “power”

Protection by using a large n