agresti/franklin statistics, 1 of 111 chapter 9 comparing two groups learn …. how to compare two...

Agresti/Franklin Statistics, 1 of 111

Chapter 9Comparing Two Groups

Learn ….How to Compare Two Groups On a Categorical or Quantitative Outcome Using Confidence Intervals and Significance Tests


Bivariate Analyses

The outcome variable is the response variable

The binary variable that specifies the groups is the explanatory variable


Bivariate Analyses

Statistical methods analyze how the outcome on the response variable depends on or is explained by the value of the explanatory variable


Independent Samples

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


Dependent Samples

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 people


Section 9.1

Categorical Response: How Can We Compare Two Proportions?


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)


Example: Aspirin, the Wonder Drug

Recent Titles of Newspaper Articles:

• “Aspirin cuts deaths after heart attack”

• “Aspirin could lower risk of ovarian cancer”

• “New study finds a daily aspirin lowers the risk of colon cancer”

• “Aspirin may lower the risk of Hodgkin’s”



The Physicians Health Study Research Group at Harvard Medical School

• Five year randomized study

• Does regular aspirin intake reduce deaths from heart disease?



Experiment:• Subjects were 22,071 male physicians

• Every other day, 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



Results displayed in a contingency table:



What is the response variable?

What are the groups to compare?



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

The groups to compare are:

• Group 1: Physicians who took a placebo

• Group 2: Physicians who took aspirin



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



008.0009.0017.0)ˆˆ(

009.011037/104ˆ

017.011034/189ˆ

21

2

1

pp

p

p

Sample Statistics:



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


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(21pp

)ˆˆ(21pp

2

22

1

11)ˆ1(ˆ)ˆ1(ˆ

n

pp

n

ppse


Confidence Interval for the Difference between Two Population Proportions

The z-score depends on the confidence level

This method requires:

• 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


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(.



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



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



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


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


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


Significance Tests Comparing Population Proportions

1. Assumptions:

Categorical response variable for two groups

Independent random samples



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



2. Hypotheses: The null hypothesis is the hypothesis of

no difference or no effect:

H0: (p1- p2) =0

• Under the presumption that p1= p2, we create a pooled estimate of the common value of p1and p2

• This pooled estimate is p̂



2. Hypotheses (continued):

Ha: (p1- p2) ≠ 0 (two-sided test)

Ha: (p1- p2) < 0 (one-sided test)

Ha: (p1- p2) > 0 (one-sided test)



3. The test statistic is:

21

21

)ˆ1(ˆ)ˆ1(ˆ0)ˆˆ(

npp

npppp

z



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

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


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



Test the Hypotheses:

H0: (p1- p2) = 0

Ha: (p1- p2) ≠ 0

• Using a significance level of 0.05• Group 1: less than 1 hr. of TV per day

• Group 2: at least 1 hr. of TV per day



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


In 2002, the median net worth was estimated as $89,000 for white households and $6000 for black households.

What is the response variable?

a. Net worth

b. Households: white or black



What is the explanatory variable?

a. Net worth

b. Households: white or black



Identify the two groups that are the categories of the explanatory variable.

a. White and Black householdsb. Net worth and households


The estimated medians were based on a sample of households. Were the samples of white households and black households independent samples or dependent samples?

a. Independent samplesb. Dependent samples



Section 9.2

Quantitative Response: How Can We Compare Two Means?


Comparing Means

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


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)



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



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



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

We estimate (µ1 - µ2) by (x1 - x2):

2.8 – 1.6 = 1.2

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



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

)(21xx


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


Confidence Interval for the Difference between Two Population Means

A 95% CI:

Software provides the t-score with right-tail probability of 0.025

2

2

2

1

2

1

025.2)(

n

s

n

stxx

1


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


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

Data as summarized by HONC scores for the two groups:

Smokers: x1 = 5.9, s1 = 3.3, n1 = 75

Ex-smokers:x2 = 1.0, s2 = 2.3, n2 = 257



Were the sample data for the two groups approximately normal?

Most likely not for Group 2 (based on the sample statistics): x2 = 1.0, s2 = 2.3)

Since the sample sizes are large, this lack of normality is not a problem



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


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


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


Significance Tests Comparing Population Means

1. Assumptions:

• Quantitative response variable for two groups

• Independent random samples



Assumptions (continued):

Approximately normal population distributions for each group• This is mainly important for small sample sizes,

and even then the two-sided test is robust to violations of this assumption



2. Hypotheses:

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

H0: (µ1- µ2) =0



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)



3. The test statistic is:

2

2

2

1

2

1

210)(

ns

nsxx

z



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

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


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



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



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



Data:



Test the hypotheses:

H0: (µ1- µ2) =0

vs.

Ha: (µ1- µ2) ≠ 0

• using a significance level of 0.05



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



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



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


What is a point estimate of µ1- µ2?

a. 18.2 – 12.9b. 32.6 – 18.1c. 6764 - 4252d. 32.6/18.2 – 18.1/12.9

How much more time do women spend on housework than men? Data is Hours per Week.

Gender: Sample Size Mean St. Dev.

Women 6764 32.6 18.2

Men 4252 18.1 12.9


What is the standard error for comparing the means?

a. 5.3b. .076c. .297d. .088



Women 6764 32.6 18.2

Men 4252 18.1 12.9


What factor causes the standard error to be small compared to the sample standard deviations for the two groups?

a. sample meansb. sample standard deviationsc. sample sizesd. genders



Women 6764 32.6 18.2

Men 4252 18.1 12.9


Section 9.3

Other Ways of Comparing Means and Comparing Proportions


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


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


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



The test statistic for H0: µ1=µ2 is:

This method has df =n1+ n2- 2

21

21

11)(

nns

xxt



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


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

2

1

ˆˆp

p


Section 9.4

How Can We Analyze Dependent Samples?


Dependent Samples

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

The observations are called matched pairs


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



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



Data:



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



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: xd


For Dependent Samples (Matched Pairs)

Mean of Differences

=

Difference of Means



The difference (x1 – x2) between the means of the two samples equals the mean xd 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



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



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

nsx

td

d



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


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


Paired-difference 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


Example: Matched Pairs Analysis for Cell Phones and Driving Study

Boxplot of the 32 difference scores



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



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



The P-value displayed in the output is 0.000

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



95% CI for µd =(µ1 - µ2):

69.5) (31.7,or

18.950.6 )32

5.52(040.26.50



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 just how large the difference must be


Section 9.5

How Can We Adjust for Effects of Other Variables?


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



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?



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



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 what to measure it



This analysis uses three variables:

• Response variable: Whether the subject has committed aggressive acts

• Explanatory variable: Level of TV watching

• Control variable: Educational level


Control Variable

A control variable is a variable that is held constant in a multivariate analysis (more than two variables)


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

Whatever association occurs cannot be due to effect of the control 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

agresti/franklin statistics, 1 of 111 chapter 9 comparing two groups learn …. how to compare two...

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