Determining Differences Between Groups

T-testsANOVA

Some background…...

Researchers want to know the characteristics of a large group (i.e. all women). IMPOSSIBLE!!

However a representative “sample” can be selected

Results are then inferred to apply to the population.

Background

This is called inferential statistics.

Comparing Means

Standard Error of the Mean= SEM

The standard deviation of a sampling distribution of means

SD divided by the square root of “n-1”

Normal Curves

Just like with scores and SD, 68% of sample means from a population fall within +1SEM of the population mean

% of Sample Means in 1 SEM

2 SEM?

Example

Mean of a population is 50 and the SEM is 5

Very likely to select a sample with a mean of 45 but very unlikely to select a sample with a mean that is 35.

Comparing Means

Most of the time researchers want to compare 2 or more samples

IE, difference in reading ability between 3rd grade boys and girls.

More background

Initial stats approach is to assume that groups are NOT different, then test statistically.

This is the NULL HYPOTHESIS“ Boys and Girls reading abilities

do not differ”“Male heights are equal to

female heights”

Comparing Means

Obviously two means from a sample will never be exactly alike.

We want to know if they are different due to chance or is there a real reason.

Comparing means

We can obtain the probability that two means will differ simply due to chance.

If the probability is small we reject the null and accept the research hypothesis: “there is a difference between male and female height”

In other words………...

If the probability is less than .05, groups are considered different

This is known as the Alpha level.

Statistical Tests

To compare two means we use a t-test

We can use the normal curve to determine the a difference between two means is likely.

Types of t-tests

Independent (unpaired): compare two different groups

Dependent (paired): same subjects in both groups. Some treatment has been applied.

Grab a representative sample

Compare Height Means and SD

Unpaired T-Test

Results

“The probability that the mean difference we observe between the male and female heights is due to chance is .034, thus we reject the null and accept the research hypothesis”

Other ways of saying it….

The probability of obtaining such an outcome is only 3.4 in 100

Our results are likely to occur by chance less than 3.4 percent of the time.

P-Value

P less than .05 is significant (reject null, accept research)

P greater than .05 is not significant (accept null, reject research)

The p value is .05 a lot of times but you may see .01, .10, or an adjusted p value sometimes as you read.

By the way………..

The significance of our example isn’t very meaningful since we probably already knew that males on average are significantly taller than females and there are reasons for our that.

Directional Hypotheses

Maybe our research hypothesis should have been: “Males are taller than females”

Before: “Male and female heights are different” (two-tailed)

Tails

One-tailed T-test We decide ahead of time that we

will subtract the mean of the females from the males (thus expect a positive difference)

If we get a negative difference it will not support our research hypothesis and we would reject the null

Quick Assignment

Using the class data spreadsheet, perform a paired (dependent) t-test for thigh vs. triceps

Assignment

Use the t-test excel spreadsheet on my web page

t-test

Assignment1

ACT1=no test prepACT2=test preparationThese are two separate sets

of subject.Perform the appropriate t-

test

Assignment2

Height1 = height in third grade

Height2 = height in sixth grade

Performed on same set of subjects

Assignment3

Attitude 1=attitude toward school at beginning of the year

Attitude 2=attitude toward school at end of the year

Performed on same group of subjects

ANOVA

Again, two typesRepeated MeasuresFactorialMore than two means to

compare

Repeated Measures (ANOVA)

Used when performing repeated measures on the same group of subjects (analogous to a dependent t-test)

I.e. changes in strength over 12 weeks of training…………..

Subjects

20 subjects have bench press 1RM assessed at 1, 4, 8 and 12 weeks

One way ANOVA used to determine changes across time

If significant, then post-hoc tests (Tukey, Scheffe)

1RM changes

1 4 8 12

“post hoc” t=tests

1 4 8 12

Wk 4 > 1 p= .03Wk 8 > 1 p = .03Wk 12 > 1 p = .01Wk 12 > 4 p = .03Wk 8 > 4 p = .02Wk 12 > 8 p = .15

Factorial ANOVA

Used when have more than two groups of subjects (analogous to independent t-test)

Comparing three different types of teaching styles on 3 different groups of 3rd graders

Higher level ANOVAS

Previous examples were one-way ANOVAs (only one independent variable)

Can have studies with multiple independent variables

Variables

Dependent and Independent Variables

Dependent – what you are measuring or interested in.

Independent – usually categorical and what remains constant.

Actually………

The training study should be a mixed factorial repeated measures ANOVA since you probably have a Control group.

Results

Time (wks)

Training

Control

Example

Two groups (control and training)

Measured strength, EMG, MMG every 4 weeks for 12 weeks

Measured strength, EMG, MMG of both legs

Measured strength, EMG, MMG at 7 different velocities

Variables?

Independent?Dependent?A 2 x 2 x 4 x 7 mixed factorial

repeated measures ANOVA for each dependent variable (otherwise would have to do a MANOVA – two or more dependent variables)

Other Higher Order Examples

You want to know how three separate third grade classes respond to 3 different teaching styles and you measured test scores at 3 different times during the year.

Answer

3 (treatment) by 3 (week of year) factorial ANOVA (3 separate groups of subjects).

Example

You are following a group of 1st graders throughout their elementary years and measuring the height of both the boys and girls.

Answer

6 (years of elementary) x 2 (gender) repeated measures (same group of subjects) mixed factorial ANOVA

Example

You are following a group of 1st graders throughout their elementary years and measuring the height (twice a year) of both the boys and girls.

Answer

6 (years of elementary) x 2 (gender) x 2 (times measured) repeated measures mixed factorial ANOVA

Columns of Data

Year1-1, Year1-2, Year 2-1, Year2-2, Year3-1……………

Would also have a gender column denoting which subjects are girls and which are boys (boys = 1, girls = 2)

Assumptions

1. Data are normally distributed2. Random sampling3. Groups are independent of

each other (random assignment into treatment groups)

4. In studies using multiple samples, the populations represented are assumed to be equally variable

Non-parametric Tests

Mann-WhitneyKruskal-WallisSign TestFriedmanChi-Square