Logic Behind Statistical Tests

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T-Test

T-test is a inferential statistical test that compares the means of two samples.

It is preferable to a z test for instances in which the real standard error is not known. Estimated standard error is used, computed from the sample variance and provides an estimate of the standard distance between a sample mean M and the population mean u.

The General Formula for t is: t = (the sample statistic - population parameter) / estimated standard error.

This holds for a Single Sample T-test.

Effect Size (Cohen's d): d = Mean Difference/Sample SD (s =

√(SS/df))

Small Effect: d=0.2; Medium Effect: d=0.5; Large Effect: d=0.8

Paired Sample T-test (within subjects design) - differences of subjects in 2 conditions.

Compares the means of two variables

Calculates case by case difference scores

Tests to see whether average difference > 0

Difference (D) of the scores of the subject in two conditions used to calculate SS for Estimated Standard Error.

Calculate

ΣD,

(ΣD)2,

ΣD2, & M for each condition

Calculate SS: SS = ΣD2 –

((ΣD)2/N)

Calculate

variance (s2): s2

=

SS/(n-1)

Calculate

estimated standard error (smD): smD

= √(s2/n)

Calculate t: t = MD/smD

d

= sample mean difference / sample standard deviation

r2

= t2 /

(t2+df) (r squared = % of variance accounted for)

Independent Samples T-test (between subjects design) - Compares means of 2 groups on one variable.

Estimated error calculated via the pooled variance between the groups.

Calculate ΣX,

ΣX2,

& M for each group

Calculate Sum of Squares (SS) for

each group

Calculate

Pooled Variance: (s2p): s2p

= (SS1

+ SS2)

/ ((n1-1)

+ (n2-1))

Estimate

standard error (s(m1-m2)): s(m1-m2)

= √((s2p /

n1)+(s2p /

n2))

Calculate

difference between the mean scores of the two groups

Look

on t-distribution table for closest df (always use the lowest of two options)

Effect

size (Cohen’s d) d

= mean difference / standard deviation SD

= √(pooled

variance)

r2

= t2 /

(t2+df)

Levene's Test for Equality of Variance: If not significant that means the variances are not statistically significant (this is good).

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One way ANOVA

In ANOVA,

the variable that designates the groups being compared is called a factor

– The individual conditions that

make up a factor are called the levels

of the factor.

•NOTE: based on variance not

mean differences Cannot compute sample mean differences

with more than two samples

Between Treatments Variance

Between-Treatments

variance: How much difference exists between

treatment conditions?

Possible explanations for differences:

•

Treatment effect:

Differences caused by the treatments

•

Chance:

Differences due to chance

• Individual

Differences

Within-Treatments variance:

Variability inside each treatment condition

Provides a measure of how much difference

is reasonable to expect by chance.

Notational

keys:

k =

the number of treatment conditions

n =

number of scores in each treatment

N =

total number of scores in the study

T =

total for each treatment condition (ΣX)

G =

sum of all the scores in the study (Grand Mean)

SS =

sum of squares (ΣX2)

df =

degrees of freedom

S2 =

variance = SS/df

Independent Measures ANOVA (Between Subjects)

Calculate ΣX, (ΣD)2, ΣD2, & M for each condition

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Factorial ANOVA

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Summary

Design

Test

Formula

SPSS

Between

Subjects Design:

Independent Samples t

(e.g. ‘Red Effect’: Adult Human Males’;

LSD Running Speed in rats)

t = (M1 – M2) / s(m1-m2)

Analyse – Compare Means – Independent

Samples t test

Within

Subjects Design:

Paired Samples t

(e.g. ‘Red Effect’: Macaques)

t = (MD - µ) / smD

Analyse – Compare Means – Paired

Samples t-test

Other:

Single Sample t

(e.g. Testing 2 Coins)

t = (M – µ) / sm

Compare means – One-Sample T Test –

Test Value is ‘specific known value’