Using Statistics in Research

Psych 231: Research Methods in Psychology

“Generic” statistical test Tests the question:

Are there differences between groups due to a treatment?

Two possibilities in the “real world”

XBXA

H0 is true (no treatment effect)

One population

Two samples

“Generic” statistical test Tests the question:

Are there differences between groups due to a treatment?

Two possibilities in the “real world”

XBXA XBXA

H0 is true (no treatment effect) H0 is false (is a treatment effect)

Two populations

Two samples

“Generic” statistical test

Why might the samples be different?(What is the source of the variability between groups)?

ER: Random sampling error ID: Individual differences (if between subjects

factor) TR: The effect of a treatment

XBXA

“Generic” statistical test

The generic test statistic - is a ratio of sources of variability

Observed difference

Difference from chance=

TR + ID + ER

ID + ER=Computed

test statistic

XBXA

ER: Random sampling error ID: Individual differences (if between

subjects factor) TR: The effect of a treatment

“Generic” statistical test

The generic test statistic distribution To reject the H0, you want a computed test statistics

that is large• This large difference, reflects a large Treatment Effect (TR)

What’s large enough? The alpha level gives us the decision criterionDistributi

on of the test statistic

-level determines where these boundaries go

“Generic” statistical test

Distribution of the test statistic

Reject H0

Fail to reject H0

The generic test statistic distribution To reject the H0, you want a computed test statistics

that is large• This large difference, reflects a large Treatment Effect (TR)

What’s large enough? The alpha level gives us the decision criterion

“Generic” statistical test

Things that affect the computed test statistic Size of the treatment effect

• The bigger the effect, the bigger the computed test statistic

Difference expected by chance (sample error)• Sample size• Variability in the population

Some inferential statistical tests

1 factor with two groups T-tests

• Between groups: 2-independent samples • Within groups: Repeated measures samples (matched, related)

1 factor with more than two groups Analysis of Variance (ANOVA) (either between groups or repeated measures)

Multi-factorial Factorial ANOVA

T-test Design

2 separate experimental conditions Degrees of freedom

• Based on the size of the sample and the kind of t-test Formula:

T =

X1 - X2Diff by chance

Based on sample error

Observed difference

Computation differs for between and within t-tests

T-test Reporting your results

The observed difference between conditions Kind of t-test Computed T-statistic Degrees of freedom for the test The “p-value” of the test

“The mean of the treatment group was 12 points higher than the control group. An independent samples t-test yielded a significant difference, t(24) = 5.67, p < 0.05.”

“The mean score of the post-test was 12 points higher than the pre-test. A repeated measures t-test demonstrated that this difference was significant significant, t(12) = 5.67, p < 0.05.”

Analysis of Variance

Designs More than two groups

• 1 Factor ANOVA, Factorial ANOVA• Both Within and Between Groups Factors

Test statistic is an F-ratio Degrees of freedom

Several to keep track of The number of them depends on the design

XBXA XC

Analysis of Variance

More than two groups Now we can’t just compute a simple difference score since there are more than one difference

So we use variance instead of simply the difference• Variance is essentially an average difference

Observed variance

Variance from chanceF-ratio =

XBXA XC

1 factor ANOVA

1 Factor, with more than two levels Now we can’t just compute a simple difference score since there are more than one difference• A - B, B - C, & A - C

XBXA XC

1 factor ANOVA

Null hypothesis: H0: all the groups are

equalXA = XB = XCAlternative hypotheses

HA: not all the groups are equal

XA ≠ XB ≠ XC

XA ≠ XB = XCXA = XB ≠

XC

XA = XC ≠ XB

The ANOVA tests this one!!

Do further tests to pick between these

XBXA XC

1 factor ANOVA

Planned contrasts and post-hoc tests:

- Further tests used to rule out the different Alternative

hypothesesXA ≠ XB ≠ XC

XA ≠ XB = XC

XA = XB ≠ XC

XA = XC ≠ XB

Test 1: A ≠ BTest 2: A ≠ CTest 3: B = C

1 factor ANOVA Reporting your results

The observed differences Kind of test Computed F-ratio Degrees of freedom for the test The “p-value” of the test Any post-hoc or planned comparison results

“The mean score of Group A was 12, Group B was 25, and Group C was 27. A 1-way ANOVA was conducted and the results yielded a significant difference, F(2,25) = 5.67, p < 0.05. Post hoc tests revealed that the differences between groups A and B and A and C were statistically reliable (respectively t(1) = 5.67, p < 0.05 & t(1) = 6.02, p <0.05). Groups B and C did not differ significantly from one another”

Factorial ANOVAs

We covered much of this in our experimental design lecture

More than one factor Factors may be within or between Overall design may be entirely within, entirely

between, or mixed Many F-ratios may be computed

An F-ratio is computed to test the main effect of each factor

An F-ratio is computed to test each of the potential interactions between the factors

Factorial ANOVAs Reporting your results

The observed differences• Because there may be a lot of these, may present them in a table instead of directly in the text

Kind of design• e.g. “2 x 2 completely between factorial design”

Computed F-ratios• May see separate paragraphs for each factor, and for interactions

Degrees of freedom for the test• Each F-ratio will have its own set of df’s

The “p-value” of the test• May want to just say “all tests were tested with an alpha level of 0.05”

Any post-hoc or planned comparison results• Typically only the theoretically interesting comparisons are presented