Inferential Statistics & Hypothesis Testing
What’s in it for me?
Okay, before your head explodes, just remember there is a pay off.
Here’s all the rest from Chapter 15 that you need to know.
Between
For data that are beyond nominal (interval or ratio; ordinal often included) you may want to see whether you can have the following effects in the dependent variable:
a) One level of a variable produces different scores
b) than another level of that variable
c) comparing the average scores of participants in the first condition to the average scores of the participants in the second
Above we have a between subjects design—two groups of participants, one in one condition one in the other. You compare the difference of the averages.
Within
You can also compare:
a) one participant to another,
b) one to him or herself at another time or under another circumstance, or
c) each subject to a standard score obtained at another time, such as a standardized test norm or something like “national average”
These are within-subject designs, sometimes referred to as
a) repeated measures for repeated scores from the same individual
b) matched pairs when there are partners such as twins or carefully selected dyads who match each other on critical characteristics
c) One-sample designs when there is one group and their scores are being compared to some kind of standard
Number of factors or variables |
Levels in factor(s) |
Research Question |
Example |
Statistical Test |
Test Statistic |
1 |
2 |
Compare means between separate groups |
Boys & girls on math scores |
Between subjects t-test Also called Independent t-test |
t |
1 |
2 |
Compare means same group at different times or under different conditions |
Children’s math scores at the beginning and end of tutoring |
Within subjects t-test Also called matched pair, repeated measures t-test |
t |
1 |
2 but only one group measured one time |
Compare one group’s score to one single benchmark |
Children’s IQ in town with toxic waste dump compared to standard mean IQ score of 100 |
One sample t-test |
t |
1 |
>2 |
Compare means |
Childen’s math scores in urban, suburban, and rural schools |
1 way ANOVA* |
F |
> 2 |
> 2 |
Compare means; main effects and interactions |
Do math scores vary by gender as well as types of math tutoring |
ANOVA **
|
F
|
*Note ANOVA described by the number of factors; n factors à n way ANOVA
**Note ANOVA may be applied to both within and between factors as well as to a combination of between and within in the same study
If you are using ordinal data or beyond (interval and ratio), and you want to see how the data for two different variables correspond to each other within individuals, you are looking for a correlation. When you want to see how one “tends to go along with” another, you could plot the pairs of datapoints on a scatterplot
Two variables |
Compare correspondence within individuals |
Children’s verbal and math scores |
Pearson Product Moment Correlation |
r |
So, all of these statistics are based on assumptions about normal distributions. As it turns out, not everything in life fits a bell curve. When your data are not normally distributed, there are other types of tests to compute. They are called non-parametric tests.
This shift from parametric (assumes normality) to non-parametric is similar to the use of medians rather than means when the distributions warrant.
What about nominal data? When you want to examine the correspondence between categories, you use tests for independence on proportions.
Two variables |
Compare correspondence across categories |
Relation between gender and dropping out of HS. Compare % of Male & Female in school to % of M-F among drop outs |
Chi-square test
|
X 2
|
What’s up with the parentheses?
Take a look at the examples on pages 450-454. Those numbers in ( ) after the test statistic give you an idea of the size of the study. They are called “degrees of freedom” and are generally N-1.
An example will illustrate the concept of “degrees of freedom.”
Think about these numbers 5, 7, 9, 3. These four give you an average of 6.
What if we wanted to create another series of four numbers that also gave you a mean of 6? What numbers would they have to be? Well, the first three could be any at all, right? It is the last one that is constrained based on the values of the first three. In other words, all but one are “free” to vary, and the final one must conform.
|
Original |
AA |
BB |
CC |
1. |
5 |
1 |
10 |
2 |
2. |
7 |
0 |
682 |
4 |
3. |
9 |
2 |
99 |
6 |
4. |
3 |
|
|
|
Mean |
6 |
6 |
6 |
6 |
So, having randomly selected three numbers for AA, BB, and CC, can the value of the fourth be any old random number? No, in order to produce the same average, its value is dictated by the first three.
What values do you fill in the yellow cells above? If you computed 21, -767, and 12, you are correct!!!!!
Still don’t know what’s in it for you? Well, for your final research proposal, you must be able to say what statistical test you will use to analyze your data. If you refer to the tables above, you should be able to do so.
You are NOT responsible for Section 15.6, Special Statistics for Research