Final Exam Review Class

8/10/2019 Final Exam Review Class

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BS704 Review for Final Exam


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Final Exam

You may bring 5 pages of notes

You MUST bring full copies ofstatistical tables (on Blackboard)

You MUST bring a calculator


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Topics Covered Since Midterm:

Hypothesis Testing for a single mean andproportion, and for two means

One-way ANOVA

Chi-square Tests

Power and Sample size

Regression and Correlation

Logistic Regression

Survival analysis


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Hypothesis Tests for...

Single mean

Single proportionp

Comparing two means 1- 2

Paired (or matched) data d


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Conducting a Hypothesis Test

Define null and research hypotheses

Define test statistic, level of

significance and decision rule

Calculate test statistic based uponsample data.

Use decision rule or p-value to decidewhether to reject or not reject the nullhypothesis.


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For a single mean

If n 30, use z-test statistic

If n < 30 use t-test statistic

For a single proportionp

Use z-test statistic

Check assumptions


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For comparing two means 1- 2

If n1and n2both 30, use z-test statistic

If n1and/or n2< 30 use t-test statistic

For comparing two proportionsp1-p2

Use chi-square test


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Type I and II errors

Type I error occurs when we reject nullhypothesis when we shouldnt.

Pr(Type I error) =

Type II error occurs when we dontreject null hypothesis when we should

have.

Pr(Type II error) =


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One-Way ANOVA

Used when we want to compare the meansof three or more groups from independentpopulations.

Continuous outcome measured on eachsubject.

We set up an analysis of variance table and

compare the variances of between groupsand within groups.

An F-test is used with two different degreesof freedom terms.


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Chi-Square Test

Chi-square goodness of fit test

Assess whether responses fit a specifieddistribution for one sample of people

Chi-square test of independence

Test if two discrete variables are associated insome way for a sample of people

Chi-square test comparing distributions

Compare distributions of proportions among twoor more independent groups


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Calculating a Sample Size for a Study

Need a large enough sample to ensureyou have the pre-specified amount of

precision in analysis Sample size determined based on type

of planned analysis:

Confidence interval

Hypothesis test


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Calculating a Sample Size for a Study

We always round up our calculation.

Need to account for possible dropout

from study. This always increases therequired sample size.


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Power

Linked up with Type II error

Power = 1-

=P(Reject H0 | H0false)

= Probability of correctlyrejecting H0when H0is false.


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Correlation

Correlation measures the nature andstrength of linear association between

two variables at a time.

Regression equation that best

describes relationship betweenvariables.


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Correlation Coefficient

Population correlation is r (rho)

Sample correlation is rwhere

-1 < r< +1

Sign indicates nature of relationship(positive or direct, negative or inverse)

Magnitude indicates strength


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Linear Regression

A very popular method for describingthe linear relationship between two

variables (usually continuousvariables).

We use a scatterplot to display the

data graphically A line to show the association between

the two variables.


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Simple Linear Regression

Y = Dependent, Outcome variable

X = Independent, Covariate, Predictor

variable

= b0+ b1x

b0is the Y-intercept, b1is the slope

y


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Multiple Linear Regression

Useful when we want to jointlyexamine the effect of several X

variables on the outcome Y variable. Y = continuous outcome variable

X1, X2, , Xp= set of independent or

predictor variables

. xb+...+xb+xb+b=y pp22110


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Linear Regression

Predictors can be continuous, indicatorvariables (0/1) or a set of dummy variables

Confoundingthe effect of a risk factor onan outcome is somehow changed due to theeffect of another factor.

Effect Modificationa different relationship

between the risk factor and an outcomedepending on the level of another variable.


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Logistic Regression

Used when the outcome is dichotomous(binary), e.g. diseased , not diseased.

Our goals remain the same as for linearregression:

is there an association between a

variable X and our outcome variable Y?If so, what type?


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Simple Logistic Regression

We model the probabilitypof havingthe disease.

Xbb

Xbb

10

10

e1

ep

xbbp1

pln)plogit( 10


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Multiple Logistic Regression

Outcome is dichotomous (1=event,0=non-event) and p=P(event)

Outcome is modeled as log odds

Exp(bi) = OR

pp22110 xb...xbxbbp-1

pln


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Survival Analysis

Outcome is the time to an event.

An event could be time to heart attack,

cancer remission or death. Measure whether person has event or not

(Yes/No) and if so, their time to event.

Determine factors associated with longersurvival.


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Survival Analysis

Incomplete follow-up information

Censoring

Measure follow-up time and not time toevent

We know survival time > follow-up time

Log rank test to compare survival intwo or more independent groups


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Cox Proportional Hazards Model

Model:

ln(h(t)/h0(t)) = b1X1+ b2X2+ + bpXp

Exp(bi) = hazard ratio

Model used to jointly assess effects of

independent variables on outcome(time to an event).


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BS704 Practice Problems forFinal Exam


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Suppose a cross-sectional study is

conducted to investigate cardiovascular riskfactors among a sample of patients seeking

medical care at one of three local hospitals.A total of 300 patients are enrolled. Usingthe following data, test if there is anassociation between enrollment site (i.e.,hospital) and family history of CVD. Runthe appropriate test at a 5% level ofsignificance.

Problem 1.


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Family

Hx

Hosp 1 Hosp 2 Hosp 3

Definite 24 14 22

Probable 8 14 8

No 68 72 70

Total 100 100 100

Problem 1.


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H0: Site and family history areindependent

H1: H

0is false =0.05

Df = (r-1)(c-1) = (3-1)(3-1) = 4.

Reject H0if 2> 9.49

Problem 1.


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FamilyHx

Hosp 1 Hosp 2 Hosp 3

Definite 24 (20) 14 (20) 22 (20)

Probable 8 (10) 14 (10) 8 (10)

No 68 (70) 72 (70) 70 (70)

Total 100 100 100

Problem 1.


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= 0.8 + 1.8 + 0.2 + 0.4 + 1.6 + 0.4 + 0.06+ 0.06 + 0 = 5.32

Do not reject H0because 5.32


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The following table summarizes data collectedin the study described in problem 1. Thevariable summarized below is body mass

index (BMI) computed as the ratio of weightin kilograms to height in meters squared.

BMI Overall Hosp 1 Hosp 2 Hosp 3

N 300 100 100 100Mean 24.8 21.6 24.8 27.9

Std Dev 2.5 2.1 1.8 1.3

Problem 2.


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Test if there is a significant difference in the mean BMI

scores among hospitals. Show all parts of the test anduse a 5% level of significance. (HINT: MSE = 3.1).

H0: 123H1: means not all equal =0.05

=100((21.6-24.8)2+(24.824.8)2+(27.924.8)2)

= 100(10.24 + 0 + 9.61) = 1985

2

jj )XX(nSSb

Problem 2.


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Source SS Df MS F

Between 1985 2 992.5 320.2

Error 920.7 297 3.1

Total 2905.7 299

Reject H0if F > 3.09

F = 320.2

Reject H0since 320.2 > 3.09. We have significantevidence, =0.05, to show that the means are notall equal.

Problem 2.


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Suppose each participant in the studydescribed in problem 1 is assigned acardiovascular risk (a value between 0 and

100 with higher scores indicative of morerisk of cardiovascular disease). The meancardiovascular risk is 21.7 with a standarddeviation of 5.6. Suppose that the

covariance between BMI and cardiovascularrisk is 4.5.

Problem 3.


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Compute the sample correlation coefficient betweenBMI and cardiovascular risk.

Var(BMI) = sx2= 2.52Var(Risk) = sy

2 = 5.62

0.3)6.5()5.2(

4.5

ss

Y)Cov(X,r222

y

2

x

Problem 3.


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2r12)(nrZ

4.5

)3.0(1

2983.0Z

2

Is this correlation statistically significant?Run the appropriate test at a 5% level of significance.

H0: r= 0H1: r0 =0.05

Reject H0if Z < -1.96 or if Z > 1.96

Reject H0since 5.4 > 1.96. We have significantevidence, =0.05, to show that r0.


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Suppose we restrict our attention to thesubgroup of patients at high risk forcardiovascular disease (cardiovascular

risk score of 30 or more).

Using the following data, test if BMI is

significantly different in men versuswomen. Use a 5% level of significance.

Problem 5.


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BMI Men Women

N 20 10

Mean 31.6 28.1

Std Dev 1.7 2.121

21

n

1

n

1Sp

XXt

H0: 1= 2H1: 1 2 =0.05

Df=20+10-2 = 28Reject H0if t < -2.048 or if t > 2.048

Problem 5.


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84.121020

)1.2(9)7.1(19 22

pS

Reject H0since 4.91>2.048. We have significant evidence,=0.05, to show there is a difference in mean BMIbetween men and women.

91.4

10

1

20

11.84

28.1-31.6t

Problem 5.


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Problem 6.

How many men and women would be required toestimate a difference in mean BMI with a 95%confidence interval and a margin of error notexceeding 1 unit. (Use data from problem 6 asneeded.)

Need 27 men and 27 women.

2

iE

Zs2n

26.011

1.96(1.84)2n

2

i

Use Sp from #6


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Problem 7.

The following table was constructed based on acomparison of various sociodemographiccharacteristics between men and women enrolled in

the study of cardiovascular risk factors.

Which, if any, of the characteristics shownabove are significantly different between men

and women? Justify.


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Problem 9.

Two different scales are used in a particularlaboratory. There is some concern that onescale gives different readings than the other.

Ten specimens are randomly selected andweighed on each scale. The data are shownbelow.

Test if there is a significant difference inweights between the two scales at =0.05


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Specimen Scale 1 Scale 2

1 1.2 2.1

2 3.5 3.6

3 1.8 1.9

4 4.0 4.0

5 5.0 4.9

6 1.9 2.0

7 2.7 2.7

8 2.2 2.3

9 2.8 2.9

10 3.5 3.7

Problem 9.


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0.1510

1.5

n

diffXd

0.276

9

/10(1.5)0.91

1n

/ndiffdiffs

222

d

H0: d= 0H1: d0 =0.05

1ndf,ns

Xtd

d Reject H0if t < -2.262 or if t > 2.262

1.72

10

0.276

0.15

ns

Xt

d

d

Do not reject H0because -2.262 < 1.72 < 2.262. We do nothave significant evidence at =0.05 to show that d0


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Patients with hypertension are generally

recommended to follow a low salt diet.Surveys report that approximately 75% of

patients adhere to these diets. In a randomsample of 100 patients with hypertension,70% report following a low-salt diet. Arethese patients significantly low in terms of

adherence? Run the test at = 0.05.

Problem 10.


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H0: p = 0.75H1: p < 0.75 =0.05

Reject H0if Z < -1.645

Do not reject H0because -1.15 > -1.645. We do nothave significant evidence at =0.05 to show that p


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Risk Factors Outcome = Systolic Blood

Pressure

RegressionCoefficient

p

Intercept 105.3 0.0001

Age

1.2

0.0042

Male Sex

4.5

0.0956

Current Smoker -0.5 0.2354

Number of Hrs

Exercise/Week

-2.4 0.0003

The following table was presented in a journal and describesthe associations between demographic and clinical riskfactors and systolic blood pressure.

Problem 11.


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a) What type of analysis generated the results summarizedabove?

Multiple linear regression analysis because the outcome(systolic blood pressure) is continuous.

b) Which of the risk factors are significantly associated withsystolic blood pressure?

Age and number of hours of exercise are statisticallysignificant at the 5% level (both have p values < 0.05). Malesex is marginally significant with a p value of 0.0956.

Problem 11.


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c) What is the relative importance of the risk factors?

The most important (statistically significant) risk factor is number ofhours of exercise per week, followed by age and then male sex.

Current smoking status is not statistically significant.

d) How would you interpret the regression coefficient associated withmale sex? With number of hours of exercise per week?

Mens systolic blood pressure is 4.5 units higher than womensholding age, smoking status and number of hours of exerciseconstant. Each additional hour of exercise per week is associatedwith a reduction of 2.4 units of systolic blood pressure holding age,sex and current smoking status constant.

Problem 11.


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Risk Factors Outcome = Hypertension

Regression Coefficient

p

Intercept 3.5 0.0001

Age

0.02

0.0357

Male Sex

0.27

0.0264

Current Smoker

-0.005

0.7564

Number of Hrs Exercise/Week -0.36 0.0111

The following table was presented in a journal and describesthe associations between demographic and clinical risk factorsand hypertension.

Problem 12.


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a) What type of analysis generated the results summarized above?

Multiple logistic regression analysis because the outcome(hypertension) is dichotomous.

b) Which of the risk factors are significantly associated withhypertension?

Age, male sex and number of hours of exercise are statisticallysignificant at the 5% level (both have p values < 0.05).

c) What is the relative importance of the risk factors?The most important (statistically significant) risk factor is number ofhours of exercise per week, followed by male sex and then age.Current smoking status is not statistically significant.

Problem 12.


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Radiation Surgery Neither Total


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H0: Age and treatment recommendation are independentH1: H0is false

=0.05

E

EO 22 )(

Df = (r-1)(c-1) = (3-1)(3-1) = 4.

Reject H0if 2> 9.49

Problem 13.

)3741()9.2830()1.3429()3750()9.2815()1.3435( 2222222


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Radiation Surgery Neither Total

9.49. We have significant evidence, =0.05,to show that age and treatment recommendation are not independent.


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Problem 14.

For each of the following scenarios,indicate which test would be used. Usethe letters below to indicate the test in

the space provided. Note that the sametest might be used for more than onescenario.


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Problem 14.

a) Compare mean to historical/external control

b) Compare proportion to historical/external control

c) Compare two independent means

d) Compare two matched/paired means

e) Analysis of variance

f) Chi-square goodness of fit test

g) Chi-square test of independence

h) Correlation analysis

i) Linear regression analysisj) Logistic regression analysis

k) Survival analysis


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Problem 14.Scenario

Test

1. We want to test if there is a significant association between BMI (kg/m2) andincident myocardial infarction adjusting for age, sex, systolic blood pressure andsmoking.

j

2. We want to test if a new environmental intervention is effective in reducingexposure to second-hand smoke. Each participant in the study has levels of exposuremeasured before and after the intervention is implemented.

d

3. We wish to test if there is a significant association between GRE scores and first

year GPA in MPH students who matriculated in fall 2011.

h or i

4. We want to determine if there are significant differences in ages of participantsenrolled in a study comparing those with a family history of cardiovascular disease tothose without.

c

5. A study reports that 15% of college freshman smoke. We want to test ifsignificantly more BU freshman smoke.

b

6. We want to test if there is a difference in preterm versus term deliveries amongwomen of black, Hispanic and white race.

g

7. We want to test if nutritional supplements prolong life (minimize time to death) in

persons over 65 years of age, adjusted for sex and other comorbid conditions.

k

8. A clinical trial is run to assess the safety of a new drug compared to a standarddrug and the outcome is development of skin rash or not

g or j

9. We want to test if there is a difference in mean time to complete a physical taskwhen comparing 12, 13, 14 and 15 year olds.

e

10. We want to test whether smoking in pregnancy increases the risk of infection innewborns.

g or k

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