1 chapter 10: comparing two groups section 10.1: categorical response: how can we compare two...

1

Chapter 10: Comparing Two Groups

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

2

Learning Objectives

1. Bivariate Analyses2. Independent Samples and Dependent Samples3. Categorical Response Variable4. Example5. Standard Error for Comparing Two Proportions6. Confidence Interval for the Difference Between

Two Population Proportions7. Interpreting a Confidence Interval for a

Difference of Proportions

3

Learning Objectives

9. Significance Tests Comparing Population Proportions

10.Examples

11.Class Exercises

4

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

5

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: Randomized experiments that

randomly allocate subjects to two treatments

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

6

Learning Objective 2:Dependent Samples

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

7

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:

(p1 – p2)

8

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

9

Results displayed in a contingency table:


10

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


11

Estimate the difference between the two population parameters of interest p1: the proportion of the population who would

have a heart attack if they participated in this experiment and took the placebo

p2: the proportion of the population who would have a heart attack if they participated in this experiment and took the aspirin


12

008.0009.0017.0)ˆˆ(

009.011037/104ˆ

017.011034/189ˆ

21

2

1

pp

p

p

Sample Statistics:


13

To make an inference about the difference of population proportions, (p1 – p2), we need to learn about the variability of the sampling distribution of: )ˆˆ(

21pp


14

Learning Objective 5:Standard Error for Comparing Two Proportions

The difference, , is obtained from sample data

It will vary from sample to sample

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

)ˆˆ(21

pp

)ˆˆ(21

pp

2

22

1

11)ˆ1(ˆ)ˆ1(ˆ

n

pp

n

ppse

15

Learning Objective 6:Confidence Interval for the Difference Between Two Population Proportions

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

2

22

1

11

21

)ˆ1(ˆ)ˆ1(ˆ)ˆˆ(

n

pp

n

ppzpp

16

Learning Objective 6:Confidence Interval Comparing Heart Attack Rates for Aspirin and Placebo 95% CI:

0.011) (0.005,or ,003.0008.011037

)009.1(009.

11034

)017.1(017.96.1)009.017(.

17

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 (p1- p2) are positive, we infer that (p1- p2) is positive

Conclusion: The population proportion of heart attacks is larger when subjects take the placebo than when they take aspirin

18

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

19

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

20

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 (p1- p2) are positive, you

can infer that (p1- p2) >0 If all values in the CI for (p1- p2) are negative,

you can infer that (p1- p2) <0 Which group is labeled ‘1’ and which is labeled

‘2’ is arbitrary

21

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

22

Learning Objective 8:Significance Tests Comparing Population Proportions

1. Assumptions:

Categorical response variable for two groups

Independent random samples

23

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


24

Learning Objective 8:Significance Tests Comparing Population Proportions2. Hypotheses: The null hypothesis is the hypothesis of no

difference or no effect:

H0: p1=p2 The alternative hypothesis is the hypothesis of

interest to the investigator

Ha: p1≠p2 (two-sided test)

Ha: p1<p2 (one-sided test)

Ha: p1>p2 (one-sided test)

25

Learning Objective 8:Significance Tests Comparing Population ProportionsPooled Estimate Under the presumption that p1= p2, we

estimate the common value of p1 and p2 by the proportion of the total sample in the 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 (n1+n2)

26


3. The test statistic is:

where is the pooled estimate

z ( ˆ p 1 ˆ p 2) 0

ˆ p (1 ˆ p )1

n1

1

n2

ˆ p

27


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 H0 and supporting Ha

28

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

29


30

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

p1 = population proportion committing aggressive acts for the lower level of TV watching

p2 = population proportion committing aggressive acts for the higher level of TV watching

31

Test the Hypotheses:

H0: (p1- p2) = 0

Ha: (p1- p2) ≠ 0

using a significance level of 0.05 Test statistic:


ˆ p 5 154

88 6190.225

se0 ˆ p 1 ˆ p 1

n1

1

n2

0.225(0.775)

1

88

1

619

0.0476

z ˆ p 1 ˆ p 2

se0

0.057 0.2490.0476

0.1920.0476

4.04

32


33

Conclusion: Since the P-value is less than 0.05, we reject H0

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


34

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:

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

Summer Status Men Women

Employed 718 593

Not Employed 79 139

Total 797 732

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

35

Null: The proportion of male students who had summer jobs is the same as the proportion of female students who had summer jobs. [H0: p1 = p2]

Alt: The proportion of male students who had summer jobs differs from the proportion of female students who had summer jobs. [Ha: p1 ≠ p2]

Hypotheses:


36

n1 = 797 and n2 = 732 (both large, so test statistic follows a Normal distribution) Pooled sample proportion:

Test statistic:

Test Statistic:


37

Hypotheses: H0: p1 = p2

Ha: p1 ≠ p2

Test Statistic:

z = 5.07 P-value:

P-value = 2P(Z > 5.07) = 0.000000396 (using a computer)

Conclusion:

Since the P-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.


38

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?

39

Learning Objective 10:Test of Significance: Two ProportionsClass 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%,

40

Learning Objective 10:Test of Significance: Two ProportionsClass 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:

favor Undecided Opposed Total

Dem 135 80 65 200

Rep 95 60 65 220

Total 230 140 130

Is there a difference in the proportions of Democrats and Republicans who are undecided regarding mandatory testing for AIDS? Test at α=5%

41


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

42

Learning Objectives

1. Comparing Means2. Standard Error for Comparing Two Means3. Confidence Interval for the Difference

between Two Population Means4. Example: Nicotine – How Much More

Addicted Are Smokers than Ex-Smokers?5. How Can We Interpret a Confidence Interval

for a Difference of Means?6. Significance Tests Comparing Population

Means

43

Learning Objective 1:Comparing Means

We can compare two groups on a quantitative response variable by comparing their means

44

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)

45

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


46

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


47

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

We estimate (µ1 - µ2) by ( ):

2.8 – 1.6 = 1.2

On average, females answered “yes” to about one more question on the HONC scale than males did


x 1 x 2

48

To make an inference about the difference between population means, (µ1 – µ2), we need to learn about the variability of the sampling distribution of:

)(21

xx


49

Learning Objective 2:Standard Error for Comparing Two Means

The difference, , is obtained from sample data. It will vary from sample to sample.

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

)xx(21

)xx(21

2

2

2

1

2

1

n

s

n

sse

50

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

A confidence interval for 1 – 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 df to be the smaller of (n1-1) and (n2-1) as a “safe” estimate

x 1 x 2 t.025

s12

n1

s2

2

n2

51

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

52

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, s1 = 3.3, n1 = 75

Ex-smokers: = 1.0, s2 = 2.3, n2 = 257

x 1

x 2

53

Were the sample data for the two groups approximately normal? Most likely not for Group 2 (based on the

sample statistics: = 1.0, s2 = 2.3) Since the sample sizes are large, this lack

of normality is not a problem


x 2

54

95% CI for (µ1- µ2):

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

)7.5 ,1.4( ,8.09.4257

3.2

75

3.3985.1)19.5(

22

or


55

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.

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

n mean std. dev.

Exercisers 29 66 8.6

Non-exercisers 31 75 9.0

56

Learning Objective 4:Example: Exercise and Pulse Rates

Find a 95% confidence interval for the difference in population means (non-exercisers minus exercisers).

Note: we use the “safe” estimate of 29-1=28 for our degrees of freedom in this calculation

“We are 95% confident that the difference in mean resting pulse rates (non-exercisers minus exercisers) is between 4.35 and 13.65 beats per minute.”

2

22

1

21

21 ns

ns

txx

75 66 2.048(9.0)2

31

(8.6)2

29

9 4.65

4.35 to 13.65

57

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 75 42.7 15.5 Non Elem. Ed 110 49.3 17.0

Find a 95% confidence interval for µ1 - µ2

58


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:

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

n Mean SD

Males 203 3.42 1.49

Females 224 2.42 1.35

59

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 (µ1 – µ2),

meaning that it is possible that µ1 = µ2

A confidence interval for (µ1 – µ2) that contains only positive numbers suggests that (µ1 – µ2) is positive We then infer that µ1 is larger than µ2

60

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

A confidence interval for (µ1 – µ2) that contains only negative numbers suggests that (µ1 – µ2) is negative We then infer that µ1 is smaller than µ2

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

61

Learning Objective 6:Significance Tests Comparing Population Means

1. Assumptions:

Quantitative response variable for two groups

Independent random samples

62


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

63


2. Hypotheses:

The null hypothesis is the hypothesis of no difference or no effect:

H0: (µ1- µ2) =0

64


2. Hypotheses (continued):

The alternative hypothesis:

Ha: (µ1- µ2) ≠ 0 (two-sided test)

Ha: (µ1- µ2) < 0 (one-sided test)

Ha: (µ1- µ2) > 0 (one-sided test)

65


3. The test statistic is:

t (x 1 x 2) 0

s12

n1

s22

n2

Note change from “z” to “t” in formula

66


4. P-value: Probability obtained from the standard normal table

5. Conclusion: Smaller P-values give stronger evidence against H0 and supporting Ha

67

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

68

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?

69

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


70

Boxplots of data:


71

Test the hypotheses:

H0: (µ1- µ2) =0

vs.

Ha: (µ1- µ2) ≠ 0

using a significance level of 0.05


72


P-Value

73

Conclusion: The P-value is less than 0.05, so we can

reject H0

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


74

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


75

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


76

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%

Mean S NFemale 2.8 3.6 150

Male 1.6 2.9 182

We would reject the claim

at a 1% level of significance

77

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

78


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

H0: µ1 - µ2 = 0, Ha: µ1 - µ2 < 0 at a level of significance equal to 0.05.

79


Section 10.3: Other Ways of Comparing Means and Comparing Proportions

80

Learning Objectives

1. Alternative Method for Comparing Means: the Pooled Standard Deviation

2. Comparing Population Means, Assuming Equal Population Standard Deviations

3. Examples

4. The Ratio of Proportions: The Relative Risk

81

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

82

Learning Objective 1:The Pooled Standard Deviation

This alternative method estimates the common value σ of σ1 and σ1 by:

2

)1()1(

21

2

22

2

11

nn

snsns

83

Learning Objective 2:Comparing Population Means, Assuming Equal Population Standard Deviations Using the pooled standard deviation estimate, a

95% CI for (µ1 - µ2) is:

This method has df =n1+ n2- 2

21

025.21

11)(

nnstxx

84

Learning Objective 2:Comparing Population Means, Assuming Equal Population Standard Deviations The test statistic for H0: µ1=µ2 is:

This method has df =n1+ n2- 2

21

21

11)(

nns

xxt

85

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

σ1=σ2

86

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

87

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

88

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.

H0: μ(nveg)= μ(veg) vs. Ha: μ(nveg)≠ μ(veg)

Mean S NNonvegetarian 3.18 1.72 51

Vegetarian 2.22 0.67 9

89


Depending on your assumption on whether the variance of both groups are equal or not impacts the conclusion of statistical significance.

Without assumption of equal variances:

90


Calculate a 95% confidence interval

Assuming Equal Variances

91

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


92

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

The ratio of proportions 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

2

1

ˆˆ

pp

93

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

Recall Physician’s Health Study:

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.

ˆ p 1 189/11034 0.0171

ˆ p 2 104 /11037 0.0094

sample relative risk = ˆ p 1 ˆ p 2 0.0171 0.0094 1.82

94


Section 10.4: How Can We Analyze Dependent Samples?

95

Learning Objectives

1. Dependent Samples

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

3. To Compare Means with Matched Pairs, Use Paired Differences

4. Confidence Interval For Dependent Samples

5. Paired Difference Inferences

96

Learning Objectives

6. Comparing Proportions with Dependent Samples

7. Confidence Interval Comparing Proportions with Matched-Pairs Data

8. McNemar’s Test

97

Learning Objective 1:Dependent Samples

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

The observations are called matched pairs

98

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

99

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


100

Data:


101

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


102

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:

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

x d

103

Learning Objective 3:To Compare Means with Matched Pairs, Use Paired Differences The difference ( – ) between the means

of the two samples equals the mean of the difference scores for the matched pairs

The difference (µ1 – µ2) between the population means is identical to the parameter µd that is the population mean of the difference scores

x 1

x 2

x d

104

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:

deviation standard their is s

sdifference theofmean sample theis

d

025.

d

dd

xn

stx

105

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

106


To test the hypothesis H0: µ1 = µ2 of equal means, we can conduct the single-sample test of H0: µd = 0 with the difference scores

The test statistic is:

1 with 0 ndf

ns

xt

d

d

107


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

108


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

109

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

110

Significance test: H0: µd = 0 (and hence equal population

means for the two conditions) Ha: µd ≠ 0

Test statistic:

46.5

325.52

6.50 t


111


112

The P-value displayed in the output is approximately 0

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


113

95% CI for µd =(µ1 - µ2):

69.5) (31.7,or

18.950.6 )32

5.52(040.26.50


114

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


115

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 833 125 958

No 2 160 162

Total 835 285 1120

116


We can estimate p1 - p2 as:

Note that the data consist of matched pairs. Recode the data so that for belief in heaven or

hell, 1=yes and 0=no

ˆ p 1 ˆ p 2 958 1120 835 1120 0.11

Heaven Hell Interpretation Difference, d Frequency

1 1 believe in Heaven and Hell 1-1=0 833

1 0 believe in Heaven, not Hell 1-0=1 125

0 1 believe in Hell, not Heaven 0-1=-1 2

0 0 do not believe in Heaven or Hell 0-0=0 160

117


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 We have converted the two samples of binary

observations into a single sample of 1120 difference scores. We can now use single-sample methods with the differences as we did for the matched-pairs analysis of means.

ˆ p 1 ˆ p 2

118

Learning Objective 7:Confidence Interval Comparing Proportions with Matched-Pairs Data Use the fact that the sample difference is

the mean of difference scores of the re-coded data

We can then find a confidence interval for the population mean of difference scores using single sample methods

ˆ p 1 ˆ p 2

119

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

n 1120

x d 0.1098

sd 0.3185

95% CI = 0.1098 1.96 0.3185 1120 0.1098 0.0187

(0.091, 0.128)

120

Learning Objective 8:McNemar Test for Comparing Proportions with Matched-Pairs Data Hypotheses: H0: p1=p2, Ha can be one or two

sided 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

121

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.

122

Learning Objective 8:Example: McNemar’s Test

Recall GSS example about belief in Heaven and Hell:

Belief in Hell

Belief in Heaven Yes No Total

Yes 833 125 958

No 2 160 162

Total 835 285 1120

123

Learning Objective 8:Example: McNemar’s Test

McNemar’s Test:

P-value is approximately 0. Note that this result agrees with the

confidence interval for p1-p2 calculated earlier

z 125 2

125 210.9

124


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

125

Learning Objectives

1. A Practically Significant Difference

2. Control Variable

3. Can An Association Be Explained by a Third Variable?

126

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

127

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?

128

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


129

We need to measure potential lurking variables and use them in the statistical analysis

If we thought that education was a potential lurking variable we would want to measure it

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

A variable that is held constant in a multivariate analysis is called a control variable


130


131

This analysis uses three variables: Response variable: Whether the

subject has committed aggressive acts

Explanatory variable: Level of TV watching

Control variable: Educational level


132

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

133

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

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

1 chapter 10: comparing two groups section 10.1: categorical response: how can we compare two...

Documents

aspirinlearning objective

subjectlearning objective

resultslearning objective

p1 p2learning objective

wonder drugexperiment

wonder drugresults

wonder drugestimate

wonder drugwhat