Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. Relationships Between Categorical Variables Chapter 6

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.

Relationships Between

Categorical Variables

Chapter 6

Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. 2

Principal Question:

Is there a relationship between the two variables, so that the category into which individuals fall for one variable seems to depend on the category they are in for the other variable?


6.1 Displaying Relationships Between Categorical Variables• Data displayed in a contingency

or two-way table.

• If one variable is explanatory, use it to define the rows of the table.

• Two types of conditional percents: row percents and column percents.

• Use row percents if the explanatory variable is the row variable.


Example 6.1 Smoking and Divorce RiskData on smoking habits and divorce history for the 1669 respondents who had ever been married.

Among smokers, 49% have been divorced, 51% have not.Among nonsmokers, only 32% have been divorced, 68% have not.The difference between row percents indicates a relationship.


Example 6.2 Tattoos and Ear PiercesResponses from n = 565 men to two questions:1. Do you have a tattoo? 2. How many total ear pierces do you have?

Among men with no ear pierces, 43/424 = 10% have a tattoo.Among men with one ear pierce, 16/70 = 23% have a tattoo.Among men with two or more ear pierces, 26/71 = 37% have a tattoo.% with a tattoo as number of ear pierces => relationshipCould examine column percents (see graph above) or overall percents too.


Example 6.3 Gender and Reasons for Taking Care of Your Body1997 poll (random-digit dialing) of 1218 southern CA residents. Question: What is the most important reason why you try to take care of your body? Is it mostly to be attractive to others, mostly to keep healthy, or mostly to help your self-confidence, or what?

Percent distribution of responses shown for men and women. Pattern of responses is very similar.Response does not seem to be related to gender.


6.2 Risk, Relative Risk, Odds Ratio, and Increased Risk

Number in category

Total number in groupRisk =

Example: Within a group of 200 individuals, asthma affects 24 people. In this group the risk of asthma is 24/200 = 0.12 or 12%.


Risk in category 1

Risk in category 2Relative Risk =

Example: For those who drive under the influence of alcohol, the relative risk of an accident is 15. => The risk of an accident for those who drive under the influence is 15 times the risk for those who don’t drive under the influence.

• Relative risk = 1 => two risks are the same.• Risk in denominator often the baseline risk.


Relative Risk of divorce = = 1.53

Example 6.1 Smoking and Divorce Risk (cont)

In this sample, the risk of divorce for smokers is 1.53 times the risk of divorce for nonsmokers.

• For smokers: risk of divorce = 238/485 = 0.491 or 49.1%.

• For nonsmokers: risk of divorce = 374/1184 = 0.316 or 31.6%.49%

32%


Difference in risks Baseline risk

Percent increase in risk

Note:When risk is smaller than baseline risk, relative risk < 1 and the percent “increase” will actually be negative, so we say percent decrease in risk.

= x 100%

= (relative risk – 1) x 100%


Relative Risk of divorce for smokers = 1.53

Example 6.1 Smoking and Divorce Risk (cont)

The risk of divorce is 53% higher for smokers than it is for nonsmokers.

Percent increase in risk of divorce for smokers = (1.53 – 1) x 100% = 53%

Difference in risks Baseline risk

= x 100% = x 100%

= 53%

(49 – 32) 32


Odds= Number in category 1 to Number in category 2 = (Number in category 1/Number in category 2) to 1

Example: Odds of getting a divorce to not getting a divorce for

smokers are 238 to 247 or 0.96 to 1.Odds of getting a divorce to not getting a divorce for

nonsmokers are 374 to 810 or 0.46 to 1.Odds Ratio = 0.96 / 0.46 = 2.1 => the odds of divorce for

smokers are about double the odds for nonsmokers.

Odds Ratio= (Odds for group 1) / (Odds for group 2)


6.3 Misleading Statistics About Risk

Questions to Ask:• What are the actual risks? What is the

baseline risk?

• What is the population for which the reported risk or relative risk applies?

• What is the time period for this risk?


Example 6.4 Disaster in the Skies? Case Study 1.2 Revisited

Look at risk of controller error per flight:

In 1998: 5.5 errors per million flights

In 1997: 4.8 errors per million flights

“Errors by air traffic controllers climbed from 746 in fiscal 1997 to 878 in fiscal 1998, an 18% increase.” USA Today

Risk of error increased but the actual risk is very small.


Example 6.5 Dietary Fat and Breast Cancer

Two reasons info is useless:

1. Don’t know how data collected nor what population the women represent.

2. Don’t know ages of women studied, so don’t know baseline rate.

“Italian scientists report that a diet rich in animal protein and fat – cheeseburgers, french fries, and ice cream, for example – increases a woman’s risk of breast cancer threefold.” Prevention Magazine’s Giant Book of Health Facts (1991, p. 122).


Example 6.5 Dietary Fat

and Breast Cancer (cont)

Age is a critical factor.Accumulated lifetime risk of woman developing breast cancer by certain ages:

By age 50: 1 in 50

By age 60: 1 in 23

By age 85: 1 in 9

Annual risk 1 in 3700 for women in early 30’s.

If Italian study was on very young women, the threefold increase in risk represents a small increase.


6.4 The Effect of a Third Variable

and Simpson’s ParadoxExample 6.7 Educational Status and

Driving after Substance Use

1996 nationwide survey of 11,847 individuals 16 or over.

Response was Driving Status with 3 categories:

• Unimpaired = never drove while impaired

• Alcohol = drove within 2 hours of alcohol use, but never after drug use

• Drug = drove within 2 hours of drug use and possibly after alcohol use.


Example 6.7 Educational Status and Driving after Substance Use

As amount of education increases, the proportion who drove within two hours of alcohol use also increases.

One difference between the educ. groups is age. Not enough information on age so age is a lurking variable.


Example 6.8 Blood Pressure and Oral Contraceptive Use

Hypothetical data on 2400 women. Recorded oral contraceptive use and if had high blood pressure.

Percent with high blood pressure is about the same among oral contraceptive users and nonusers.


Example 6.8 Blood Pressure and Oral Contraceptive Use (cont)

Many factors affect blood pressure. If users and nonusers differ with respect to such a factor, the factor confounds the results. Blood pressure increases with age and users tend to be younger.

In each age group, the percentage with high blood pressure is higher for users than for nonusers => Simpson’s Paradox.


6.5 Assessing the Statistical Significance of a 2x2 Table

Question: Can a relationship observed in the sample data be inferred to hold in the population represented by the data?

A statistically significant relationship or difference is one that is large enough to be unlikely to have occurred in the observed sample if there is no relationship or difference in the population.


Five Steps to Determining Statistical Significance:

1. Determine the null and alternative hypotheses.

2. Verify necessary data conditions, and if met, summarize the data into an appropriate test statistic.

3. Assuming the null hypothesis is true, find the p-value.

4. Decide whether or not the result is statistically significant based on the p-value.

5. Report the conclusion in the context of the situation.


Step 1: Null and Alternative Hypotheses

null hypothesis: The two variables are not related.

alternative hypotheses: The two variables are related.


Step 2: The Chi-square StatisticChi-square statistic measures the difference between the observed counts and the counts that would be expected if there were no relationship.

Large difference => evidence of a relationship.

• Compute expected count for each cell:

Expected count = (Row total)(Column total) Total n for table

• Compute for each cell: (Obs count – Exp count)2

Exp count

• Compute test statistic by totaling over all cells:

(Obs count – Exp count)2

Exp count 2


Step 3: The p-value of the Chi-square Test

Q: If there is actually no relationship in the population, what is the likelihood that the chi-square statistic could be as large as it is or larger?

A: The p-value

Large test statistic => evidence of a relationship.So how large is enough to declare significance?

Note: The p-value is generally reported in computer output.


Steps 4 and 5: Making andReporting a Decision

Common rule:• p-value 0.05 => say relationship is statistically

significant and we reject the null hypothesis

• p-value > 0.05 => cannot say relationship is statistically significant and we cannot reject the null hypothesis

Large test statistic => small p-value => evidence a real relationship exists in the population.

Note: For 22 tables, a test statistic of 3.84 or larger is significant.


Example 6.10 Randomly Pick S or Q

Of 92 college students asked: “Randomly choose one of the letters S or Q”, 66% (61/92) picked S. Of another 98 students asked: “Randomly choose one of the letters Q or S”, 46% (45/98) picked S.

Can we conclude order of letters on the form and the response are related?

The p-value = 0.005 which is less than 0.05, so the relationship is statistically significant.


Factors that Affect Statistical Significance

• The strength of the observed relationship

Example 6.10Of those with “S or Q”, 66% picked S.

Of those with “Q or S”, 46% picked S.

Difference in percentages (66% - 46%) reflects the strength of the observed relationship.


Factors that Affect Statistical Significance: (cont)

• How many people were studied

Example:I. Treatment A had 8 of 10 patients improve.

Treatment B had 5 of 10 patients improve. Strength = 80% - 50% = 30% seems large but study is too small. The p-value is 0.16.

II. Treatment A had 80 of 100 patients improve. Treatment B had 50 of 100 patients improve. Strength = 80% - 50% = 30% is again large. The p-value is 0.000000087, which is very significant.


Practical versus Statistical SignificanceStatistical Significance does not mean the

relationship is of practical importance.

Example 6.12 Aspirin and Heart Attacksp-value is 0.000 => relationship is statistically significant.

Placebo: 189/11034 = 1.71% had attack

Aspirin: 104/11037 = 0.94% had attack

Difference only 1.71 – 0.94 = 0.77%, or less than 1%.

With large sample this important difference was detected.


Interpreting a Nonsignificant Result

• The sample results are not strong enough to safely conclude that there is a relationship in the population.

• The observed relationship could have resulted by chance, even if there is no relationship in the population. This is not the same as saying there is no relationship.


Case Study 6.1 Drinking, Driving, and the Supreme Court

“Random Roadside Survey” of drivers under 20 years of age.

p-value is 0.201 => the observed association could easily have occurred even if there is no relationship in the population.

This result was used by Supreme Court to overturn a law that allowed sale of beer to females but not males.

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Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc. Relationships Between Categorical Variables Chapter 6