Psychology 320: Psychological Statistics

Professor: Howard B. Lee

Lecture Notes

Week 10 : Chapter 8 and 10

Lecture 22

Dependent T-tests

The null hypothesis and the alternative hypothesis are stated similarly whether or not you have 2 independent groups or 2 dependent groups.

How can you tell if you are dealing with dependent or independent groups?
By how the data is collected. Key words to look for which indicate that you have dependent groups are: "matched", "paired", and "correlated".

decision rule: df = n-1, where n = # of pairs

For exploratory research when you don't know what will happen, use a two-tailed test with an alpha level = .05

Critical values:

It is easier to reject ho with a one-tailed test than with a two-tailed test when you have the same alpha level.

The word "significant" usually means that you have rejected ho.

Lecture 24

Analysis of Variance (ANOVA)

IV ----> Dosage

One Way ANOVA

One independent variable (IV) with more than two levels (more than 2 groups). See Chapter 10 in your book.

Two Way ANOVA

Two independent variables (simultaneous) with each having two or more levels. See Chapter 11 in your book.

5 levels (5 groups)
# T-tests comparing each of the 5 groups? = 10.
1 vs. 2, 1 vs 3, 1 vs 4, 1 vs 5, 2 vs 3, 2 vs. 4, 2 vs. 5, 3 vs. 4, 3 vs. 5, and 4 vs. 5

(# groups)(# groups-1)/2 ==> # of T-tests
If we had 10 T-tests at alpha = .05,

or 40% chance of a type 1 error occurring in one or all of those 10 T-tests!

The way you get around this problem is by using an ANOVA.

ANOVA --> alpha = .05
overall alpha = .05

Lets consider the price you pay :

Multiple T-tests

Disadvantages	Advantages
1. Lots of computations	1. If significance exists, you know where the difference is
2. Family rate error

ANOVA

Disadvantages	Advantages
1. If significance exists, you do not know where the difference is	1. You don't have to worry about Family rate error
	2. If not significant, all computations are done. One stat. computation vs. many ( multiple T-tests)

In ANOVA, if ho is not rejected, we know there are no significant differences between the individual groups.
In ANOVA, if ho is rejected, then we follow up with multiple comparison tests to find out which groups are different.

Lecture 25

Analysis of Variance (ANOVA):

--> a more general approach.
Total variance --> The variability of the dependent variable ignoring group membership.

Total variance --> break down into components (partition).
In the one-way ANOVA for independent samples (not related to each other in any way).
Between Group Variance
Within Group Variance
Independent Variable (IV) --> Dependent Variable (DV) : Establish a cause and effect relationship
Similar groups before treatment with between group variance that changes due to treatment.
You want groups to be similar to begin with, administer dosage, then measure their performance on a certain task.
If they are different after treatment, you can say that the IV was what caused that difference.
What is wrong here?
2 groups are very different to begin with.
2 groups with similar mean weights should have been used. (ex. group 1: M = 150, group 2: M = 150)
Groups (inside) will not be completely the same, the variability before treatment is not exactly the same but not radically different either. It can be done by:

• random sampling
• matched samples

Within group variance --> same before treatment and after treatment.
Total variance = Between + Within
If there is a treatment effect, the between group variance should outweigh the within group variance.

How ANOVA is related to analysis of means:
If between group variance is large, then the means of the groups (levels of the IV) will be very different.
Ex. M = 100, M = 100, M = 100.
Between group variance = 0 or no variability at all between group means.
Ex. M = 100, M = 150, M = 200.
Between group variance = not 0.

If the means of group means change after treatment the between group variance reflects the difference in group means.

Hypothesis testing with ANOVA:

1. ho: u1 = u2 = u3
2. h1: u1 not = u2 or u1 not = u3 or u2 not = u3 or any two or any three ( at least one).
3. test statistic
4. decision rule, with alpha = .05 or alpha = .01

Standard deviation and variance will never be less than 0 so you know you always will have a right-tailed test with ANOVAs.

df between
df within
Use table E, p. 408-410.
Follows an F distribution

The null hypothesis is always written with the population mean "u", never with the sample mean "M".

For multiple groups or more than 3, this is easier to state.
h1; ui not = uj for some i,j
i = or greater than 1, j = or greater than i + 1.

Lecture 26

One Way ANOVA:

• will tell you if there is a difference or not.
• will not tell you where the difference lies.

Set up an ANOVA summary table:

source	degrees of freedom	sum of squares	mean of squares	F
between grps. (treatment)	# levels for IV - 1	SSbetween
within grps. (residual or error)	total # of subjects - # levels of IV
Total	total # of subjects - 1	SStotal

* Reminder: There are no negative numbers in an ANOVA table. On the quiz, if you see any negative numbers on the ANOVA table, you know that there is something wrong!

To calculate SStotal, use calculator.
Total Sum of Squares = SStotal

To calculate the SStotal with the calculator:

1. Put all #s in and get S.D.
2. Square the S.D. (SD2)
3. Multiply by N ( the total # of scores)

Use calculator to find S.D. for SStotal.
Hit "Mean" button to get mean.
Square it and multiply it by N.

To find SSwithin, subtract SSbetween from SStotal.

Mean Square:

MSbetween = SSbetween/df between

MSwithin = SSwithin/dfwithin

F = MSbetween/MSwithin
You must round this calculation to 2 decimal places. This F is your test statistic.