Section 8.2

Transcript Section 8.2

Chapter 10: Comparing Two Groups

Section 10.1: Categorical Response: How Can We Compare Two Proportions?

Learning Objectives

Bivariate Analyses Independent Samples and Dependent Samples Categorical Response Variable Example Standard Error for Comparing Two Proportions Confidence Interval for the Difference B etween Two Population Proportions Interpreting a Confidence Interval for a Difference of Proportions 2

Learning Objectives

Significance Tests Comparing Population Proportions

10.

Examples

11.

Class Exercises

Learning Objective 1: Bivariate Analyses

Methods for comparing two groups are special cases of bivariate statistical methods: there are two variables



The outcome variable on which comparisons are made is the response variable

 

The binary variable that specifies the groups is the

explanatory variable

Statistical methods analyze how the outcome on the response variable

depends on

or is

explained by

the value of the explanatory variable 4

Learning Objective 2 : Independent Samples

Most comparisons of groups use independent samples from the groups:



The observations in one sample are

independent

of those in the other sample



Example: randomly allocate subjects to two treatments Randomized experiments that



Example: An observational study that separates subjects into groups according to their value for an explanatory variable

Learning Objective 2 : Dependent Samples 

Dependent samples result when the data are matched pairs sample is matched with a subject in the other sample – each subject in one



Example: set of married couples, the men being in one sample and the women in the other.



Example: Each subject is observed at two times, so the two samples have the same subject

Learning Objective 3: Categorical Response Variable

For a categorical response variable



Inferences compare groups in terms of their population proportions in a particular

category



We can compare the groups by the difference in their population proportions

(p 1 – p 2 )

Learning Objective 4: Example: Aspirin, the Wonder Drug 

Experiment:



Subjects were 22,071 male physicians



Every other day for five years, study participants took either an aspirin or a placebo



The physicians were randomly assigned to the aspirin or to the placebo group



The study was double-blind: the physicians did not know which pill they were taking, nor did those who evaluated the results

Learning Objective 4: Example: Aspirin, the Wonder Drug

Results displayed in a contingency table:

Learning Objective 4: Example: Aspirin, the Wonder Drug 

What is the response variable?



The response variable is whether the subject had a heart attack, with categories ‘yes’ or ‘no’



What are the groups to compare?



The groups to compare are:



Group 1: Physicians who took a placebo



Group 2: Physicians who took aspirin

Learning Objective 4: Example: Aspirin, the Wonder Drug 

Estimate the difference between the two population parameters of interest



p 1 : the proportion of the population who would have a heart attack if they participated in this experiment and took the

placebo



p 2 : the proportion of the population who would have a heart attack if they participated in this experiment and took the

aspirin

Learning Objective 4: Example: Aspirin, the Wonder Drug

Sample Statistics:

(

1 

189 / 11034



0 .

017

2  1 

104 ) / 11037



0 .

017



0 .

009



0 .

009



0 .

008

Learning Objective 4: Example: Aspirin, the Wonder Drug 

To make an inference about the difference of population proportions, (p 1 – p 2 ), we need to learn about the variability of the sampling distribution of:

(

ˆ 1  ˆ

2 ) 13

Learning Objective 5: Standard Error for Comparing Two Proportions  ( ˆ

1 

ˆ 2 )

data



It will vary from sample to sample



This variation is the standard error of the sampling distribution of :

1 2 )

 ˆ 1 ( 1  ˆ 1 ) 

1 ˆ 2 ( 1  ˆ 2 )

2 14

Learning Objective 6: Confidence Interval for the Difference Between Two Population Proportions ( ˆ 1  ˆ 2 ) 

ˆ 1 ( 1 

1 ˆ 1 )  ˆ 2 ( 1  ˆ 2 )

2  

The z-score depends on the confidence level This method requires:



Categorical response variable for two groups



Independent random samples for the two groups



Large enough sample sizes so that there are at least 10 “successes” and at least 10 “failures” in each group

Learning Objective 6: Confidence Interval Comparing Heart Attack Rates for Aspirin and Placebo 

95% CI

: (.

017  .

009 )  1 .

96 .

017 ( 1  .

017 11034 0 .

008  0 .

003 , or (0.005, 0.011) )  .

009 ( 1  .

009 ) 11037  16

Learning Objective 6: Confidence Interval Comparing Heart Attack Rates for Aspirin and Placebo 

Since both endpoints of the confidence interval (0.005, 0.011) for (p 1 - p 2 ) are positive, we infer that (p 1 - p 2 ) is positive



Conclusion: The population proportion of heart attacks is

larger

when subjects take the placebo than when they take aspirin

Learning Objective 6: Confidence Interval Comparing Heart Attack Rates for Aspirin and Placebo  

The population difference (0.005, 0.011) is small Even though it is a small difference, it may be important in public health terms



For example, a decrease of 0.01 over a 5 year period in the proportion of people suffering heart attacks would mean 2 million fewer people having heart attacks

Learning Objective 6: Confidence Interval Comparing Heart Attack Rates for Aspirin and Placebo 

The study used male doctors in the U.S



The inference applies to the U.S. population of male doctors



Before concluding that aspirin benefits a larger population, we’d want to see results of studies with more diverse groups

Learning Objective 7: Interpreting a Confidence Interval for a Difference of Proportions     

Check whether 0 falls in the CI If so, it is plausible that the population proportions are equal If all values in the CI for (p 1 - p 2 ) are positive , you can infer that (p 1 - p 2 ) >0 If all values in the CI for (p 1 - p 2 ) are negative , you can infer that (p 1 - p 2 ) <0 Which group is labeled ‘1’ and which is labeled ‘2’ is arbitrary

Learning Objective 7: Interpreting a Confidence Interval for a Difference of Proportions 

The magnitude of values in the confidence interval tells you how large any true difference is



If all values in the confidence interval are near 0 , the true difference may be relatively small in practical terms

Learning Objective 8: Significance Tests Comparing Population Proportions

1. Assumptions:



Categorical response variable for two groups



Independent random samples

Learning Objective 8: Significance Tests Comparing Population Proportions

Assumptions (continued):



Significance tests comparing proportions use the sample size guideline from confidence intervals: Each sample should have at least about 10 “successes” and 10 “failures”



Two –sided tests are robust against violations of this condition



At least 5 “successes” and 5 “failures” is adequate

Learning Objective 8: Significance Tests Comparing Population Proportions

2. Hypotheses:



The null hypothesis is the hypothesis of no difference or no effect: H 0 : p 1 =p 2 The alternative hypothesis is the hypothesis of interest to the investigator H a : p 1 ≠p 2 (two-sided test) H a : p 1

p 2 (one-sided test)

Learning Objective 8: Significance Tests Comparing Population Proportions •

Pooled Estimate



Under the presumption that p 1 = p 2 , we estimate the common value of p 1 and p 2 the proportion of the total sample in the by category of interest This pooled estimate is calculated by combining the number of successes in the two groups and dividing by the combined sample size (n 1 +n 2 )

  Learning Objective 8: Significance Tests Comparing Population Proportions

3. The test statistic is:

 ( ˆ 1 ˆ (1   ˆ 2 ) ˆ )   1

1   0 1

2   26

Learning Objective 8: Significance Tests Comparing Population Proportions

4. P-value: Probability obtained from the standard normal table of values even more extreme than observed z test statistic 5. Conclusion: Smaller P-values give stronger evidence against H supporting H a 0 and

Learning Objective 9: Example: Is TV Watching Associated with Aggressive Behavior?



Various studies have examined a link between TV violence and aggressive behavior by those who watch a lot of TV



A study sampled 707 families in two counties in New York state and made follow-up observations over 17 years



The data shows levels of TV watching along with incidents of aggressive acts

Learning Objective 9: Example: Is TV Watching Associated with Aggressive Behavior?

 Define Group 1 as those who watched less than 1 hour of TV per day, on the average, as teenagers  Define Group 2 as those who averaged at least 1 hour of TV per day, as teenagers 

1 = population proportion committing aggressive acts for the lower level of TV watching 

p 2

= population proportion committing aggressive acts for the higher level of TV watching 30

 Learning Objective 9: Example: Is TV Watching Associated with Aggressive Behavior?

 Test the Hypotheses: H 0 : (p 1 - p 2 ) = 0 H a : (p 1 - p 2 ) ≠ 0 using a significance level of 0.05

 Test statistic: ˆ  5  154 88  619  0.225

0  ˆ  1     1

1  1

2  

 ˆ 1 

0 ˆ 2 0.225(0.775)  1 88  1 619   0.057

 0.249

0.0476

  0.192

0.0476

 4.04

0.0476

Learning Objective 9: Example: Is TV Watching Associated with Aggressive Behavior?

 

Conclusion: Since the P-value is less than 0.05, we reject H 0 We conclude that the population proportions of aggressive acts differ for the two groups



The sample values suggest that the population proportion is higher for the higher level of TV watching

Learning Objective 9: Test of Significance: Two Proportions  Summer Jobs Example A university financial aid office polled a simple random sample of undergraduate students to study their summer employment.

 Not all students were employed the previous summer. Here are the results:

Summer Status

Employed Not Employed

Total Men

718 79 797

Women

593 139 732  Is there evidence that the proportion of male students who had summer jobs differs from the proportion of female students who had summer jobs ?

Learning Objective 9: Test of Significance: Two Proportions Summer Jobs Example

Hypotheses:

 Null: The proportion of male students who had summer jobs is the same as the proportion of female students who had summer jobs. [H 0 :

1 =

2 ]  Alt: The proportion of male students who had summer jobs differs from the proportion of female students who had summer jobs. [H a :

1 ≠

2 ] 35

Learning Objective 9: Test of Significance: Two Proportions Summer Jobs Example

Test Statistic :



1 = 797 and

2 = 732 (both large, so test statistic follows a Normal distribution )  Pooled sample proportion:  Test statistic: 36

 Learning Objective 9: Test of Significance: Two Proportions Summer Jobs Example

Hypotheses:

H 0 :

1 H a :

1 =

2 ≠

2  

Test Statistic:

z = 5.07

P-value:

-value = 2

(

> 5.07) = 0.000000396

(using a computer)



Conclusion:

Since the

-value is quite small, there is very strong evidence that the proportion of male students who had summer jobs differs from that of female students.

Learning Objective 9: Test of Significance: Two Proportions Drinking and unplanned sex   In a study of binge drinking, the percent who said they had engaged in unplanned sex because of drinking was 19.2% out of 12708 in 1993 and 21.3% out of 8783 in 2001 Is this change statistically significant at the 0.05 significance level?

The P-value is 0.0002 < .05. The results are statistically significant. But are they practically significant?

Learning Objective 10: Test of Significance: Two Proportions Class Exercise 1  A survey of one hundred male and one hundred female high school seniors showed that thirty-five percent of the males and twenty-nine percent of the females had used marijuana previously. Does this survey indicate a difference in proportions for the population of high school seniors? Test at α=5%, 39

Learning Objective 10: Test of Significance: Two Proportions Class Exercise 2  A random sample of 500 persons were questioned regarding political affiliation and attitude toward government sponsored mandatory testing of AIDS. The results were as follows: Dem Rep favor 135 95 Undecided 80 60 Total 230 140 Opposed 65 65 130 Total 200 220 Is there a difference in the proportions of Democrats and Republicans who are undecided regarding mandatory testing for AIDS? Test at α=5% 40

Chapter 10: Comparing Two Groups

Section10.2: Quantitative Response: How Can We Compare Two Means?

Learning Objectives

Comparing Means Standard Error for Comparing Two Means Confidence Interval for the Difference between Two Population Means Example: Nicotine – How Much More Addicted Are Smokers than Ex-Smokers?

How Can We Interpret a Confidence Interval for a Difference of Means?

Significance Tests Comparing Population Means 42

Learning Objective 1 : Comparing Means 

We can compare two groups on a

quantitative response variable

comparing their means by

Learning Objective 1: Example: Teenagers Hooked on Nicotine 

A 30-month study:



Evaluated the degree of addiction that teenagers form to nicotine



332 students who had used nicotine were evaluated



The response variable was constructed using a questionnaire called the Hooked on Nicotine Checklist (HONC)

Learning Objective 1: Example: Teenagers Hooked on Nicotine 

The HONC score is the total number of questions to which a student answered “yes” during the study



The higher the score, the more hooked on nicotine a student is judged to be

Learning Objective 1: Example: Teenagers Hooked on Nicotine 

The study considered explanatory variables, such as gender, that might be associated with the HONC score

Learning Objective 1: Example: Teenagers Hooked on Nicotine 

How can we compare the sample HONC scores for females and males?



We estimate (µ 1 µ 2

1 

2.8 – 1.6 = 1.2



about one more question on the HONC scale than males did

Learning Objective 1: Example: Teenagers Hooked on Nicotine 

To make an inference about the difference between population means, (µ 1 – µ 2 ), we need to learn about the variability of the sampling distribution of:

(

1 

2 ) 48

Learning Objective 2 : Standard Error for Comparing Two Means  ( x 1  x 2 )

data. It will vary from sample to sample.



This variation is the standard error of the

( x 1  x 2 )



1 2

1 

2 2

2 49

Learning Objective 3: Confidence Interval for the Difference Between Two Population Means A

confidence interval for



1 

2  

.025

1 2

1 

2 2

–

is:    t .025

is the critical value for a 95% confidence level from the t distribution The

degrees of freedom

are calculated using software. If you are not using software, you can take

to be the smaller of (

1 -1) and (

2 -1) as a “safe” estimate 50

Learning Objective 3 : Confidence Interval for the Difference between Two Population Means 

This method assumes

: 

Independent random samples from the two groups



An approximately normal population distribution for each group



this is mainly important for small sample sizes, and even then the method is robust to violations of this assumption

Learning Objective 4: Example: Nicotine – How Much More Addicted Are Smokers than Ex-Smokers?



Data as summarized by HONC scores for the two groups:



Smokers: = 5.9, s 1 = 3.3, n 1 = 75



Ex-smokers: = 1.0, s 2 = 2.3, n 2 = 257

  52

Learning Objective 4: Example: Nicotine – How Much More Addicted Are Smokers than Ex-Smokers?



Were the sample data for the two groups approximately normal?

 

Most likely not for Group 2 (based on the

2 = 2.3) Since the sample sizes are large, this lack of normality is not a problem

 53

Learning Objective 4: Example: Nicotine – How Much More Addicted Are Smokers than Ex-Smokers?



95% CI for (µ 1 µ 2 ):

( 5 .

9



1 )



1 .

985 3 .

3 75 4 .

9



0 .

8 ,

( 4 .

1 , 5 .

7 )

2 

2 .

3

257

 

We can infer that the population mean for the smokers is between 4.1 higher and 5.7 higher than for the ex-smokers

Learning Objective 4: Example: Exercise and Pulse Rates A study is performed to compare the mean resting pulse rate of adult subjects who exercise regularly to the mean resting pulse rate of those who do not exercise regularly.

Exercisers Non-exercisers

29 31 mean 66 75 std. dev.

8.6

9.0

This is an example of when to use the two-sample t procedures.

Learning Objective 4: Example: Exercise and Pulse Rates Find a 95% confidence interval for the difference in population means (non-exercisers minus exercisers).

1 

2 



2 1

1 

2 2

2   75 9   66 4.65

 2.048

(9.0)2 31  (8.6)2 29 4.35 to 13.65

 Note: we use the “safe” estimate of 29-1=28 for our degrees of  rates (non-exercisers minus exercisers) is between 4.35 and 13.65 beats per minute.” 56

Learning Objective

4:

Class Exercise 1

 Attitude toward mathematics was measured for two different groups. The attitude scores range from 0 to 80 with the higher scores indicating a more positive attitude. The first group consisted of Elementary education majors and the other group consisted of majors from several other areas. The results were as follows: N mean SD   Elementary Ed Non Elem. Ed 75 42.7

110 49.3

15.5

17.0

 Find a 95% confidence interval for µ 1 µ 2 57

Learning Objective 4

:

Class Exercise 2

 Are girls less inclined to enroll in science courses than boys? One recent study of fourth, fifth, and sixth graders asked how many science courses they intended to take. The resulting data were used to compute the following summary statistics: Males Females

203 224

Mean

3.42

2.42

1.49

1.35

 Calculate a 99% confidence interval for the difference between males and females in mean number of science courses planned 58

Learning Objective 5: How Can We Interpret a Confidence Interval for a Difference of Means?

 

Check whether 0 falls in the interval When it does, 0 is a plausible value for (µ 1 µ 2 ), meaning that it is possible that µ 1 = µ 2 –



A confidence interval for (µ 1 – µ 2 ) that contains only positive numbers suggests that (µ 1 – µ 2 ) is positive



We then infer that µ 1 is larger than µ 2

Learning Objective 5: How Can We Interpret a Confidence Interval for a Difference of Means?



A confidence interval for (µ 1 – µ 2 ) that contains only negative numbers suggests that (µ 1 – µ 2 ) is negative



We then infer that µ 1 is smaller than µ 2



Which group is labeled ‘1’ and which is labeled ‘2’ is arbitrary

Learning Objective 6: Significance Tests Comparing Population Means

1. Assumptions:



Quantitative response variable for two groups



Independent random samples

Learning Objective 6: Significance Tests Comparing Population Means

Assumptions (continued):



Approximately normal population distributions for each group



This is mainly important for small sample sizes, and even then the two-sided t test is robust to violations of this assumption

Learning Objective 6: Significance Tests Comparing Population Means

2. Hypotheses: The null hypothesis is the hypothesis of

no difference

no effect

: H 0 : (µ 1 µ 2 ) =0

Learning Objective 6: Significance Tests Comparing Population Proportions

2. Hypotheses (continued): The alternative hypothesis: H a : (µ 1 H a : (µ 1 H a : (µ 1 µ 2 ) ≠ 0 (two-sided test) µ 2 ) < 0 (one-sided test) µ 2 ) > 0 (one-sided test)

Learning Objective 6: Significance Tests Comparing Population Means

3. The test statistic is:

 (

1 

1 2

2 )  0 

s n

2 2 2 Note change from “z” to “t” in formula  65

Learning Objective 6: Significance Tests Comparing Population Means

4. P-value: Probability obtained from the standard normal table 5. Conclusion: Smaller P-values give stronger evidence against H 0 supporting H a and

Learning Objective 6: Example: Does Cell Phone Use While Driving Impair Reaction Times?



Experiment:



64 college students



32 were randomly assigned to the cell phone group



32 to the control group

Learning Objective 6: Example: Does Cell Phone Use While Driving Impair Reaction Times?



Experiment (continued):



Students used a machine that simulated driving situations



At irregular periods a target flashed red or green



Participants were instructed to press a “brake button” as soon as possible when they detected a red light

Learning Objective 6: Example: Does Cell Phone Use While Driving Impair Reaction Times?



For each subject, the experiment analyzed their mean response time over all the trials



Averaged over all trials and subjects, the

mean

response time for the cell-phone group was 585.2

milliseconds



The

mean

response time for the control group was 533.7

milliseconds

Learning Objective 6: Example: Does Cell Phone Use While Driving Impair Reaction Times?



Boxplots of data:

Learning Objective 6: Example: Does Cell Phone Use While Driving Impair Reaction Times?



Test the hypotheses: H 0 : (µ 1 µ 2 ) =0 vs.

H a : (µ 1 µ 2 ) ≠ 0



using a significance level of 0.05

Learning Objective 6: Example: Does Cell Phone Use While Driving Impair Reaction Times?

P-Value 72

Learning Objective 6: Example: Does Cell Phone Use While Driving Impair Reaction Times?



Conclusion:



The P-value is less than 0.05, so we can reject H 0



There is enough evidence to conclude that the population mean response times differ between the cell phone and control groups



The sample means suggest that the population mean is higher for the cell phone group

Learning Objective 6: Example: Does Cell Phone Use While Driving Impair Reaction Times?



What do the box plots tell us?



There is an extreme outlier for the cell phone group



It is a good idea to make sure the results of the analysis aren’t affected too strongly by that single observation



Delete the extreme outlier and redo the analysis



In this example, the t-statistic changes only slightly

Learning Objective 6: Example: Does Cell Phone Use While Driving Impair Reaction Times?



Insight:



In practice, you should not delete outliers from a data set without sufficient cause (i.e., if it seems the observation was incorrectly recorded)



It is however, a good idea to check for sensitivity of an analysis to an outlier



If the results change much, it means that the inference including the outlier is on shaky ground

Learning Objective 6: Example: Females or males more nicotine dependent  Test the claim that there is a difference between males and females and their level of dependence on nicotine with a level of significance of 1% Female Male Mean 2.8

1.6

S 3.6

2.9

N 150 182 We would reject the claim at a 1% level of significance 76

Learning Objective

6:

Class exercise 1

 Many people take ginkgo supplements advertised to improve memory. Are these over the counter supplements effective?

 Based on the study results below, is there evidence that taking 40 mg of ginkgo 3 times a day is effective in increasing mean performance?

 Test the relevant hypothesis using α=5%

n Mean S

Ginkgo 104 5.6

0.6

Placebo 115 5.5

0.6

Learning Objective

6:

Class Exercise 2

   Attitude toward mathematics was measured for two different groups. The attitude scores range from 0 to 80 with the higher scores indicating a more positive attitude. One group consisted of Elementary education majors and the other group consisted of majors from several other areas. The results were as follows: N mean SD Elementary Ed 75 42.7

15.5

Non Elem. Ed 110 49.3

17.0

 Calculate the P-value, and give your conclusion for testing H 0 : µ 1 0.05. µ 2 = 0, H a : µ 1 µ 2 < 0 at a level of significance equal to 78

Chapter 10: Comparing Two Groups

Section 10.3: Other Ways of Comparing Means and Comparing Proportions 79

Learning Objectives 1.

Alternative Method for Comparing Means : the Pooled Standard Deviation Comparing Population Means, Assuming Equal Population Standard Deviations Examples The Ratio of Proportions: The Relative Risk 80

Learning Objective 1: Alternative Method for Comparing Means 

An alternative t- method can be used when, under the null hypothesis, it is reasonable to expect the

variability

as well as the mean to be the same



This method requires the assumption that the population standard deviations be equal

Learning Objective 1: The Pooled Standard Deviation 

This alternative method estimates the common value σ of σ 1 and σ 1 by:



(

1 

1 )

1 2

1  

(

2 

2



1 )

2 2 82

Learning Objective 2: Comparing Population Means, Assuming Equal Population Standard Deviations 

Using the pooled standard deviation estimate, a 95% CI for (µ 1 µ 2 ) is:

(

1 

)



025

1

1 

1

2 

This method has df =n 1 + n 2 - 2

Learning Objective 2: Comparing Population Means, Assuming Equal Population Standard Deviations 

The test statistic for H 0 : µ 1 =µ 2

 (

1 1 



is:

1 ) 1

2 

This method has df =n 1 + n 2 - 2

Learning Objective 2: Comparing Population Means, Assuming Equal Population Standard Deviations 

These methods assume:



Independent random samples from the two groups



An approximately normal population distribution for each group



This is mainly important for small sample sizes, and even then, the CI and the two-sided test are usually robust to violations of this assumption



σ 1 = σ 2

Learning Objective 3: Example: Is Arthroscopic Surgery better than Placebo?

 Calculate the P-Value and determine if there is a statistical difference between Arthroscopic surgery and Placebo at 5% level of significance.

With a P-value of 0.63, we should not reject the null that there is no difference between placebo and Arthroscopic surgery 86

Learning Objective 3: Example: Is Arthroscopic Surgery better than Placebo?

 Calculate a 95% Confidence Interval   We are 95% Confident that the difference between the placebo and surgery is in this range -10.6 to 6.4.

Notice that 0 is within this range. Thus, we should not reject the null hypothesis at the 5% significance level that there is no difference between the two treatment groups 87

Learning Objective 3 : Example: Are Vegetarians More Liberal?

  Respondents were rated on a scale of 1-7 with 1 being liberal and 7 being the most conservative. Is there a significant difference between Non-vegetarian and vegetarians? Assume equal variances.

H 0 : μ (nveg) = μ (veg) vs. H a : μ (nveg) ≠ μ (veg) Nonvegetarian Vegetarian Mean 3.18

2.22

S 1.72

0.67

N 51 9 88

Learning Objective 3: Example: Are Vegetarians More Liberal?

Without assumption of equal variances: Depending on your assumption on whether the variance of both groups are equal or not impacts the conclusion of statistical significance. 89

Learning Objective 3: Example: Are Vegetarians More Liberal?

 Calculate a 95% confidence interval Assuming Equal Variances 90

Learning Objective 3: Example: Are Vegetarians More Liberal?

 Assuming unequal variances, what is the 95% Confidence Interval?

Learning Objective 4: The Ratio of Proportions: The Relative Risk 

The

ratio of proportions

1 ˆ

for two groups is:

 

In medical applications for which the proportion refers to a category that is an undesirable outcome, such as death or having a heart attack, this ratio is called the relative risk The ratio describes the sizes of the proportions relative to each other

Learning Objective 4: The Ratio of Proportions: The Relative Risk  Recall Physician’s Health Study:   ˆ 1 ˆ 2  189 /11034  104 /11037   0.0171

0.0094

sample relative risk = ˆ 1 ˆ 2  0.0171 0.0094

 1.82

The proportion of the placebo group who had a heart attack was 1.82 times the proportion of the aspirin group who had a heart attack.

Chapter 10: Comparing Two Groups

Section 10.4: How Can We Analyze Dependent Samples?

Learning Objectives

Dependent Samples Example: Matched Pairs Design for Cell Phones and Driving Study To Compare Means with Matched Pairs, Use Paired Differences Confidence Interval For Dependent Samples Paired Difference Inferences 95

Learning Objectives

Comparing Proportions with Dependent Samples Confidence Interval Comparing Proportions with Matched-Pairs Data McNemar’s Test 96

Learning Objective 1: Dependent Samples 

Each observation in one sample has a matched observation in the other sample



The observations are called

matched pairs

Learning Objective 2: Example: Matched Pairs Design for Cell Phones and Driving Study 

The cell phone analysis presented earlier in this text used independent samples:



One group used cell phones



A separate control group did not use cell phones

Learning Objective 2: Example: Matched Pairs Design for Cell Phones and Driving Study 

An alternative design used the same subjects for both groups



Reaction times are measured when subjects performed the driving task without using cell phones and then again while using cell phones

Learning Objective 2: Example: Matched Pairs Design for Cell Phones and Driving Study

Data:

100

Learning Objective 2: Example: Matched Pairs Design for Cell Phones and Driving Study 

Benefits of using dependent samples (matched pairs):



Many sources of potential bias are controlled so we can make a more accurate comparison



Using matched pairs keeps many other factors fixed that could affect the analysis



Often this results in the benefit of smaller standard errors

101

Learning Objective 3: To Compare Means with Matched Pairs, Use Paired Differences 

To Compare Means with Matched Pairs, Use Paired Differences:



For each matched pair, construct a difference score



d = (reaction time using cell phone) – (reaction time without cell phone)



Calculate the sample mean of these differences:

x d

 102

Learning Objective 3: To Compare Means with Matched Pairs , Use Paired Differences  

The difference (

–

) between the means

x d

the difference scores for the matched pairs

 

parameter µ d 1 – µ 2 ) between the of the difference scores



that is the population mean

103

Learning Objective 4: Confidence Interval For Dependent Samples 

Let n denote the number of observations in each sample



This equals the number of difference scores



The 95 % CI for the population mean difference is:

x d



025

s d n x d

is the sample mean of the difference s s

is their standard deviation

104

Learning Objective 5 : Paired Difference Inferences 

These

paired-difference inferences

are special cases of single-sample inferences about a population mean so they make the same assumptions

105

Learning Objective 5 : Paired Difference Inferences  

To test the hypothesis H 0 : µ 1 = µ 2 of equal means, we can conduct the single-sample test of H 0 : µ d = 0 with the difference scores The test statistic is:



x d s d

 0 with

df n



 1 106

Learning Objective 5 : Paired Difference Inferences 

Assumptions:



The sample of difference scores is a random sample from a population of such difference scores



The difference scores have a population distribution that is approximately normal



This is mainly important for small samples (less than about 30) and for one-sided inferences

107

Learning Objective 5 : Paired Differ ence Inferences 

Confidence intervals and two-sided tests are

robust

: They work quite well even if the normality assumption is violated



One-sided tests do not work well when the sample size is small and the distribution of differences is highly skewed

108

Learning Objective 5: Example: Cell Phones and Driving Study 

The box plot shows skew to the right for the difference scores



Two-sided inference is robust to violations of the assumption of normality



The box plot does not show any severe outliers

109

Learning Objective 5: Example: Cell Phones and Driving Study 

Significance test:



H 0 : µ d = 0 (and hence equal population means for the two conditions)



H a : µ d ≠ 0



Test statistic:

 50 52 .

5 .

6 32  5 .

46 110

Learning Objective 5: Example: Cell Phones and Driving Study 111

Learning Objective 5: Example: Cell Phones and Driving Study 

The P-value displayed in the output is approximately 0



There is extremely strong evidence that the population mean reaction times are different

112

Learning Objective 5: Example: Cell Phones and Driving Study 

95% CI for µ d =(µ 1 µ 2 ):

50 .

6



2 .

040 ( 52 .

5 ) 32



50.6



18.9 or (31.7, 69.5)

113

Learning Objective 5: Example: Cell Phones and Driving Study 

We infer that the population mean when using cell phones is between about 32 and 70 milliseconds higher than when not using cell phones



The confidence interval is more informative than the significance test, since it predicts possible values for the difference

114

Learning Objective 6: Comparing Proportions with Dependent Samples  A recent GSS asked subjects whether they believed in Heaven and whether they believed in Hell:

Belief in Hell Belief in Heaven

Yes No

Total

Yes 833 2

835

No 125 160

285 Total 958 162 1120

115

Learning Objective 6: Comparing Proportions with Dependent Samples   We can estimate

1  ˆ 2

1 -

2 as:  958 1120  835 1120  0.11

 Note that the data consist of matched pairs.  Recode the data so that for belief in heaven or hell, 1=yes and 0=no

Heaven

1 1 0 0

Hell

1 0 1 0

Interpretation

believe in Heaven and Hell believe in Heaven, not Hell believe in Hell, not Heaven do not believe in Heaven or Hell

Difference, d

1-1=0 1-0=1 0-1=-1 0-0=0

Frequency

833 125 2 160 116

Learning Objective 6: Comparing Proportions with Dependent Samples  Sample mean of the 1120 difference scores is [0(833)+1(125)-1(2)+0(160)]/1120=0.11

 Note that this equals the difference in proportions 

1  ˆ 2 observations into a single sample of 1120 difference scores. We can now use single sample methods with the differences as we  117

Learning Objective 7: Confidence Interval Comparing Proportions with Matched-Pairs Data  the mean of difference scores of the re-coded data  population mean of difference scores using single sample methods 118

 Learning Objective 7: Confidence Interval Comparing Proportions with Matched-Pairs Data 

 1120

x d

 0.1098

s d

 0.3185

95% CI = 0.1098

   0.1098

 0.0187

 (0.091, 0.128)  119

Learning Objective 8: McNemar Test for Comparing Proportions with Matched-Pairs Data  Hypotheses: H 0 :

1 =

2 , H a sided can be one or two  Test Statistic: For the two counts for the frequency of “yes” on one response and “no” on the other, the z test statistic equals their difference divided by the square root of their sum.  P-value: The probability of observing a sample even more extreme than the observed sample 120

Learning Objective 8: McNemar Test for Comparing Proportions with Matched-Pairs Data  Assumptions:  The sum of the counts used in the test should be at least 30, but in practice, the two-sided test works well even if this is not true.

121

Learning Objective 8: Example: McNemar’s Test  Recall GSS example about belief in Heaven and Hell:

Belief in Heaven

Yes No

Total Belief in Hell

Yes No

Total

833 2

835

125 160

285 958 162 1120

122

Learning Objective 8: Example: McNemar’s Test  McNemar’s Test:

 125  2 125  2  10.9

P-value is approximately 0.

 confidence interval for

1 -

2 calculated earlier 123

Chapter 10: Comparing Two Groups

Section 10.5: How Can We Adjust for Effects of Other Variables?

124

Learning Objectives

A Practically Significant Difference Control Variable Can An Association Be Explained by a Third Variable?

125

Learning Objective 1: A Practically Significant Difference 

When we find a practically significant difference between two groups, can we identify a reason for the difference?



Warning: An association may be due to a lurking variable not measured in the study

126

Learning Objective 2: Control Variable 

In a previous example, we saw that teenagers who watch more TV have a tendency later in life to commit more aggressive acts



Could there be a lurking variable that influences this association?

127

Learning Objective 2: Control Variable 

Perhaps teenagers who watch more TV tend to attain lower educational levels and perhaps lower education tends to be associated with higher levels of aggression

128

Learning Objective 2: Control Variable 

We need to measure

potential lurking variables

and use them in the statistical analysis

 

Including a potential lurking variable in the study changes it from a bivariate study to a multivariate study



If we thought that education was a

potential

lurking variable we would want to measure it A variable that is held constant in a multivariate analysis is called a control

variable

129

Learning Objective 2: Control Variable 130

Learning Objective 2: Control Variable 

This analysis uses three variables:



Response variable: Whether the subject has committed aggressive acts



Explanatory variable: Level of TV watching



Control variable

: Educational level

131

Learning Objective 3: Can An Association Be Explained by a Third Variable?



Treat the third variable as a control variable



Conduct the ordinary bivariate analysis while holding that control variable constant at fixed values (multivariate analysis)



Whatever association occurs cannot be due to the effect of the control variable

132

Learning Objective 3: Can An Association Be Explained by a Third Variable?



At each educational level, the percentage committing an aggressive act is higher for those who watched more TV



For this hypothetical data, the association observed between TV watching and aggressive acts was not because of education

133