Statistical Power (Overview)

This week we'll be covering statistical power (also known as power analysis). Power is not a statistical technique like correlation, t-test, and chi-square. Rather, power involves designing your study (particularly getting a large enough sample size) so that you can use correlations, t-tests, etc., more effectively. The core concept of power, like so much else, goes back to the distinction between the population and a sample. When there truly is a basis in the population for rejecting the null hypothesis (e.g., a non-zero correlation, a non-zero difference between means), we want to increase the likelihood that we reject the null from the analysis of our sample. In other words, we want to be able to pronounce a result significant, when warranted. Here are links to my previous entries on statistical power.

Introductory lecture

Why a powerful design is needed: The population may truly have a non-zero correlation, for example, but due to random sampling error, your sample may not; plus, some songs on statistical power!

Remember that there's also the opposite kind of error: The population truly has absolutely no correlation, but again due to random sampling error, you draw a sample that gives the impression of a non-zero correlation.

How to plan a study using power considerations

Chi-Square

My introductory stat notes for methods class have some introductory information on chi-square.

Here are direct links to some old chi-square blog postings. This one discusses the reversibility error and how properly to read an SPSS printout of a chi-square analysis. The other one illustrates the null hypothesis for chi-square analyses in terms of equal pie-charts.

The following photo of the board, containing chi-square tips, was added on November 15, 2011 (thanks to Selen).

Plus a song (added November 1, 2011):

One Degree is Free

Lyrics by Alan Reifman

(May be sung to the tune of “Rock and Roll is Free,” Ben Harper)

Look at your, chi-square table,

If it is, 2-by-2,

One cell can be filled freely,

While the others take their cue,

The formula that you can use,

Come on, from the columns, lose one,

And one, as well, from the rows,

Multiply the two, isn’t this fun?

One degree is free, in your table,

With con-tin-gen-cy, in your table,

One degree is free, in your table,

…free in your table,

…free in your table,

Say, your table is larger,

Maybe it’s 2-by-4,

Multiply one by three,

3 df are in store,

The df’s are essential,

To check significance,

Go to your chi-square table,

And find the right instance,

Three degrees are free, in your table,

With con-tin-gen-cy, in your table,

Three degrees are free, in your table,

…free in your table,

…free in your table,

(Guitar Solo)

Correlation

UPDATED 10/2/2022 

Our next topic is correlational analysis. There are four major areas to address:

1. A general introduction to correlation. Correlation refers to how two variables go together ("co-relate"). There are three main types of correlations:

Positive correlation: As one variable goes up, so does the other. They both follow the same pattern. Knowing where a person stands on one variable, you know roughly where he/she stands on the other (maximum = +1.0). 

• Example: The more hours one studies before a test, the higher the score he or she will likely get. 

Zero correlation: Knowing where a person stands on one variable tells us nothing about where he/she stands on the other. Someone who has a high score on one variable is equally likely to have a high or a low value on the other. 

• Example: A person’s number of sneezes per week is (probably) uncorrelated with the percent of a person’s shirts that are blue. 

Negative correlation: As one goes up, the other goes down. They follow an inverse pattern. Knowing where a person stands on one variable, you again know roughly where he/she stands on the other (minimum = -1.0). 

• Example: The higher the winter temperatures where one lives (e.g., Miami), the fewer the heavy jackets people buy. 

Graphical depictions of positive, zero, and negative correlations. Note that the correlation (symbolized r) is based upon the slope of the best-fitting line (line which comes closest to all the points) and degree to which points are close to the line vs. being scattered. 

A song to nail down our understanding of correlation, best-fit lines, upward and downward slopes, etc. 

Fitting the Line 

Lyrics by Alan Reifman 

(May be sung to the tune of “Draggin’ the Line,” James/King) 

Plotting the data, on X and Y, 

Finding the slope, with most points nearby, 

We want to find the angle, of the trend’s incline, 

Fitting the line (fitting the line), 

Upward slopes make r positive, 

Slopes trending down, make it negative, 

From minus-one to plus-one, r can feel fine, 

Fitting the line (fitting the line), 

Fitting the line (fitting the line), 

Points align, how will the data shine? 

If you have upward slopes, it’ll give you a plus sign, 

Fitting the line (fitting the line), 

Fitting the line (fitting the line), 

How strongly will your variables relate? 

Is there a trend, or just a zero flat state? 

You want to know what your analysis will find, 

Fitting the line (fitting the line), 

Fitting the line (fitting the line), 

Points align, how will the data shine? 

Your r will be minus, if the slope declines, 

Fitting the line (fitting the line), 

Fitting the line (fitting the line), 

(Guitar solo) 

Points align, how will the data shine? 

If you have upward slopes, it’ll give you a plus sign, 

Fitting the line (fitting the line), 

Fitting the line (fitting the line)… 

Facebook album of Dr. Reifman meeting singer Tommy James, after the latter's concert at the 2013 South Plains Fair. 

2. Running correlations in SPSS. This graphic of SPSS output tries to make clear that a sample correlation and its significance/probability level are two different things (although related to each other).

Second, in graphing the data points and best-fitting line, you start in "Graphs," go to "Legacy Dialogs," and select "Scatter/Dot." Then, select "Simple Scatter" and click on "Define." You will then insert the variables you want to display on the X and Y axes, and say "OK." When the scatter plot first appears, you can click on it to do more editing. To add the best-fit line, under "Elements," choose "Fit Line at Total."

Initially the dots will all look the same throughout the scatter plot. To make each dot represent the number of cases at that point (either by thickness of the dot or through color-coding), click on the "Binning" icon (circled below in red). Thanks to Xiaohui for finding this!

3. Statistical significance and testing the null hypothesis, as applied to correlation. Subthemes within this topic include how sample size affects the ease of getting a statistically significant result (i.e., rejecting the null hypothesis of zero correlation in the full population), and one- vs. two-tailed significance. 

4. Partial correlation (i.e., the correlation between two variables, holding constant one or more "lurking" variables).

Here are some additional tips:

5. In evaluating the meaning of a correlation that appears as positive or negative in the SPSS output, you must know how each of the variables is keyed (i.e., does a high score reflect more of the behavior or less of the behavior?).

6. Statistical significance is not necessarily indicative of social importance. With really large sample sizes (such as we have available in the GSS), even a correlation that seems only modestly different from zero may be statistically significant. To remedy this situation, the late statistician Jacob Cohen devised criteria for "small," "medium," and "large" correlations.

7. Correlations should also be interpreted in the context of range restriction (see links section on the right). Here's a song to reinforce the ideas:

Restriction in the Range 

Lyrics by Alan Reifman

(May be sung to the tune of “Laughter in the Rain,” Sedaka/Cody)

Why do you get such a small correlation,
With variables you think should be related?
Seems you’re not studying the full human spectrum,
Just looking at part of bivariate space,
All kinds of thoughts start to race, through your mind…

Ooh, there’s restriction in the range,
Dampening the slope of the best-fit line,
Ooh, I can correct r for this,
Put a better rho estimate in its place...