CPH Exam Review Biostatistics Lisa Sullivan, PhD Associate Dean for Education Professor and Chair, Department of Biostatistics Boston University School of Public Health

Outline and Goals  Overview of Biostatistics (Core Area)  Terminology and Definitions  Practice Questions An archived version of this review, along with the PPT file, will be available on the NBPHE website ( under Study Resourceswww.nbphe.org

Biostatistics Two Areas of Applied Biostatistics: Descriptive Statistics Summarize a sample selected from a population Inferential Statistics Make inferences about population parameters based on sample statistics.

Variable Types  Dichotomous variables have 2 possible responses (e.g., Yes/No)  Ordinal and categorical variables have more than two responses and responses are ordered and unordered, respectively  Continuous (or measurement) variables assume in theory any values between a theoretical minimum and maximum

We want to study whether individuals over 45 years are at greater risk of diabetes than those 45 and younger. What kind of variable is age? 1.Dichotomous 2.Ordinal 3.Categorical 4.Continuous

We are interested in assessing disparities in infant morbidity by race/ethnicity. What kind of variable is race/ethnicity? 1.Dichotomous 2.Ordinal 3.Categorical 4.Continuous

Numerical Summaries of Dichotomous, Categorical and Ordinal Variables Frequency Distribution Table Heath StatusFreq.Rel. Freq.Cumulative Freq Cumulative Rel. Freq. Excellent1938%1938% Very Good1224%3162% Good918%4080% Fair612%4692% Poor48%50100% n=50100% Ordinal variables only

Frequency Bar Chart

Relative Frequency Histogram

Continuous Variables  Assume, in theory, any value between a theoretical minimum and maximum  Quantitative, measurement variables  Example – systolic blood pressure

Summarizing Location and Variability  When there are no outliers, the sample mean and standard deviation summarize location and variability  When there are outliers, the median and interquartile range (IQR) summarize location and variability, where IQR = Q 3 -Q 1  Outliers Q IQR

Mean Vs. Median

Box and Whisker Plot Min Q1 Median Q3 Max

Comparing Samples with Box and Whisker Plots Systolic Blood Pressure 1 2

What type of display is shown below? 1.Frequency bar chart 2.Relative frequency bar chart 3.Frequency histogram 4.Relative frequency histogram Percent Patients by Disease Stage

The distribution of SBP in men, years is shown below. What is the best summary of a typical value 1.Mean 2.Median 3.Interquartile range 4.Standard Deviation

When data are skewed, the mean is higher than the median. 1.True 2.False

The best summary of variability for the following continuous variable is 1.Mean 2.Median 3.Interquartile range 4.Standard Deviation

Numerical and Graphical Summaries

What is the probability of selecting a male with optimal blood pressure? 1.20/ / /150 Blood Pressure Category Optimal Normal Pre-Htn Htn Total Male Female Total

What is the probability of selecting a patient with Pre-Htn or Htn? 1.95/ / /150 Blood Pressure Category Optimal Normal Pre-Htn Htn Total Male Female Total

What proportion of men have prevalent CVD? CVD Free of CVD Men Women / / /300

What proportion of patients with CVD are men ? CVD Free of CVD Men Women / / /300

Are Family History and Current Status Independent? Example. Consider the following table which cross classifies subjects by their family history of CVD and current (prevalent) CVD status. Current CVD Family HistoryNoYes No21525 Yes9015 P(Current CVD| Family Hx) = 15/105 = P(Current CVD| No Family Hx) = 25/240 = 0.104

Are symptoms independent of disease? Disease No Disease Total Symptoms No Symptoms No 2.Yes

Popular Probability Models – Discrete Outcomes BinomialPoisson OutcomeSuccess/FailureCount Number of response categories 2>2 Number of trials/replications FixedInfinite Relationships among trials Independent

Probability Models – Normal Distribution  Model for continuous outcome  Mean=median=mode

Normal Distribution Properties of Normal Distribution I) The normal distribution is symmetric about the mean (i.e., P(X > ) = P(X < ) = 0.5). ii) The mean and variance ( and  2 ) completely characterize the normal distribution. iii) The mean = the median = the mode iv) Approximately 68% of obs between mean + 1 sd 95% between mean + 2 sd, and >99% between mean + 3 sd

Normal Distribution Body mass index (BMI) for men age 60 is normally distributed with a mean of 29 and standard deviation of 6. What is the probability that a male has BMI < 29? P(X<29)= 0.5

Normal Distribution P(X<30)=? What is the probability that a male has BMI less than 30?

Standard Normal Distribution Z Normal distribution with =0 and =

Normal Distribution P(X<30)= P(Z<0.17) = From a table of standard normal probabilities or statistical computing package.

Comparing Systolic Blood Pressure (SBP) Comparing systolic blood pressure (SBP)  Suppose for Males Age 50, SBP is approximately normally distributed with a mean of 108 and a standard deviation of 14  Suppose for Females Age 50, SBP is approximately normally distributed with a mean of 100 and a standard deviation of 8 If a Male Age 50 has a SBP = 140 and a Female Age 50 has a SBP = 120, who has the “relatively” higher SBP ?

Normal Distribution Z M = ( ) / 14 = 2.29 Z F = ( ) / 8 = 2.50 Which is more extreme?

Percentiles of the Normal Distribution The k th percentile is defined as the score that holds k percent of the scores below it. Eg., 90 th percentile is the score that holds 90% of the scores below it. Q1 = 25 th percentile, median = 50 th percentile, Q3 = 75 th percentile

Percentiles For the normal distribution, the following is used to compute percentiles: X =  + Z  where  = mean of the random variable X,  = standard deviation, and Z = value from the standard normal distribution for the desired percentile (e.g., 95 th, Z=1.645). 95 th percentile of BMI for Men: (6) = 38.9

Central Limit Theorem  (Non-normal) population with   Take samples of size n – as long as n is sufficiently large (usually n > 30 suffices)  The distribution of the sample mean is approximately normal, therefore can use Z to compute probabilities Standard error

Statistical Inference  There are two broad areas of statistical inference, estimation and hypothesis testing.  Estimation. Population parameter is unknown, sample statistics are used to generate estimates.  Hypothesis Testing. A statement is made about parameter, sample statistics support or refute statement.

What Analysis To Do When  Nature of primary outcome variable Continuous, dichotomous, categorical, time to event  Number of comparison groups One, 2 independent, 2 matched or paired, > 2  Associations between variables Regression analysis

Estimation  Process of determining likely values for unknown population parameter  Point estimate is best single-valued estimate for parameter  Confidence interval is range of values for parameter: point estimate + margin of error point estimate + t SE (point estimate)

Hypothesis Testing Procedures 1. Set up null and research hypotheses, select  2. Select test statistic 3.Set up decision rule 4. Compute test statistic 5. Draw conclusion & summarize significance (p-value)

P-values  P-values represent the exact significance of the data  Estimate p-values when rejecting H 0 to summarize significance of the data (approximate with statistical tables, exact value with computing package)  If p <  then reject H 0

Errors in Hypothesis Tests Conclusion of Statistical Test Do Not Reject H 0 Reject H 0 H 0 trueCorrectType I error H 0 false Type II error Correct

Continuous Outcome Confidence Interval for   Continuous outcome - 1 Sample n > 30 n < 30 Statistical computing packages use t throughout.

New Scenario  Outcome is dichotomous Result of surgery (success, failure) Cancer remission (yes/no)  One study sample  Data On each participant, measure outcome (yes/no) n, x=# positive responses,

Dichotomous Outcome Confidence Interval for p  Dichotomous outcome - 1 Sample Example. In the Framingham Offspring Study (n=3532), 1219 patients were on antihypertensive medications. Generate 95% CI (0.329, 0.361)

One Sample Procedures – Comparisons with Historical/External Control Continuous Dichotomous H 0 :  0 H 0 : pp 0 H 1 :  0, < 0, ≠ 0 H 1 : pp 0, <p 0, ≠p 0 n>30 n<30 Statistical computing packages use t throughout.

One Sample Procedures – Comparisons with Historical/External Control Categorical or Ordinal outcome  2 Goodness of fit test H 0 : p 1 p 10, p 2 p 20,..., p k p k0 H 1 : H 0 is false

New Scenario  Outcome is continuous SBP, Weight, cholesterol  Two independent study samples  Data On each participant, identify group and measure outcome

Two Independent Samples Cohort Study - Set of Subjects Who Meet Study Inclusion Criteria Group 1Group 2 Mean Group 1Mean Group 2

Two Independent Samples RCT: Set of Subjects Who Meet Study Eligibility Criteria Randomize Treatment 1Treatment 2 Mean Trt 1Mean Trt 2

Continuous Outcome Confidence Interval for (      Continuous outcome - 2 Independent Samples n 1 >30 and n 2 >30 n 1 <30 or n 2 <30 Statistical computing packages use t throughout.

Hypothesis Testing for (      Continuous outcome  2 Independent Sample H 0 :    2 (   2 = 0) H 1 :    2,   < 2,   ≠ 2

Hypothesis Testing for (     Test Statistic n 1 >30 and n 2 > 30 n 1 <30 or n 2 <30 Statistical computing packages use t throughout.

An RCT is planned to show the efficacy of a new drug vs. placebo to lower total cholesterol. What are the hypotheses? 1.H 0 :  P = N H 1 :  P > N 2.H 0 :  P = N H 1 :  P < N 3.H 0 :  P = N H 1 :  P ≠ N

New Scenario  Outcome is dichotomous Result of surgery (success, failure) Cancer remission (yes/no)  Two independent study samples  Data On each participant, identify group and measure outcome (yes/no)

Dichotomous Outcome Confidence Interval for (p  p    Dichotomous outcome - 2 Independent Samples

Measures of Effect for Dichotomous Outcomes  Outcome = dichotomous (Y/N or 0/1) Risk=proportion of successes = x/n Odds=ratio of successes to failures=x/(n-x)

Measures of Effect for Dichotomous Outcomes  Risk Difference =  Relative Risk =  Odds Ratio =

Confidence Intervals for Relative Risk (RR)  Dichotomous outcome  2 Independent Samples exp(lower limit), exp(upper limit)

Confidence Intervals for Odds Ratio (OR)  Dichotomous outcome  2 Independent Samples exp(lower limit), exp(upper limit)

Hypothesis Testing for (p 1 -p 2 )  Dichotomous outcome  2 Independent Sample H 0 : p 1 =p 2 H 1 : p 1 >p 2, p 1 <p 2, p 1 ≠p 2 Test Statistic

Two (Independent) Group Comparisons Difference in birth weight is -106 g, 95% CI for difference in mean Birth weight: ( to -36.7)

New Scenario  Outcome is continuous SBP, Weight, cholesterol  Two matched study samples  Data On each participant, measure outcome under each experimental condition Compute differences (D=X 1 -X 2 )

Two Dependent/Matched Samples Subject IDMeasure 1Measure Measures taken serially in time or under different experimental conditions

Crossover Trial Treatment Eligible R ParticipantsPlacebo Each participant measured on Treatment and placebo

Confidence Intervals for  d  Continuous outcome  2 Matched/Paired Samples n > 30 n < 30 Statistical computing packages use t throughout.

Hypothesis Testing for  d  Continuous outcome  2 Matched/Paired Samples H 0 :  d  H 1 :  d ,  d <0,  d ≠0 Test Statistic n>30 n<30

Independent Vs Matched Design

Statistical Significance versus Effect Size  P-value summarizes significance  Confidence intervals give magnitude of effect (If null value is included in CI, then no statistical significance)

The null value of a difference in means is…

The null value of a mean difference is…

The null value of a relative risk is…

The null value of a difference in proportions is…

The null value of an odds ratio is…

A two sided test for the equality of means produces p=0.20. Reject H 0 ? 1.Yes 2.No 3.Maybe

Hypothesis Testing for More than 2 Means - Analysis of Variance  Continuous outcome  k Independent Samples, k > 2 H 0 :    2  …  k H 1 : Means are not all equal Test Statistic F is ratio of between group variation to within group variation (error)

ANOVA Table Source of Sums ofMean Variation SquaresdfSquaresF Between Treatmentsk-1SSB/k-1 MSB/MSE ErrorN-kSSE/N-k TotalN-1

ANOVA  When the sample sizes are equal, the design is said to be balanced  Balanced designs give greatest power and are more robust to violations of the normality assumption

Extensions  Multiple Comparison Procedures – Used to test for specific differences in means after rejecting equality of all means (e.g., Tukey, Scheffe)  Higher-Order ANOVA - Tests for differences in means as a function of several factors

Extensions  Repeated Measures ANOVA - Tests for differences in means when there are multiple measurements in the same participants (e.g., measures taken serially in time)

 2 Test of Independence  Dichotomous, ordinal or categorical outcome  2 or More Samples H 0 : The distribution of the outcome is independent of the groups H 1 : H 0 is false Test Statistic

 2 Test of Independence  Data organization (r by c table)  Is the distribution of the outcome different (associated with) groups Outcome Group123 A20%40% B50%25% C90%5%

What Tests Were Used?

In Framingham Heart Study, we want to assess risk factors for Impaired Glucose  Outcome = Glucose Category Diabetes (glucose > 126), Impaired Fasting Glucose (glucose ), Normal Glucose  Risk Factors Sex Age BMI (normal weight, overweight, obese) Genetics

What test would be used to assess whether sex is associated with Glucose Category? 1.ANOVA 2.Chi-Square GOF 3.Chi-Square test of independence 4.Test for equality of means 5.Other

What test would be used to assess whether age is associated with Glucose Category? 1.ANOVA 2.Chi-Square GOF 3.Chi-Square test of independence 4.Test for equality of means 5.Other

What test would be used to assess whether BMI is associated with Glucose Category? 1.ANOVA 2.Chi-Square GOF 3.Chi-Square test of independence 4.Test for equality of means 5.Other

In Framingham Heart Study, we want to assess risk factors for Glucose Level  Consider a Secondary Outcome = Fasting Glucose Level  Risk Factors Sex Age BMI (normal weight, overweight, obese) Genetics

What test would be used to assess whether sex is associated with Glucose Level? 1.ANOVA 2.Chi-Square GOF 3.Chi-Square test of independence 4.Test for equality of means 5.Other

What test would be used to assess whether BMI is associated with Glucose Level? 1.ANOVA 2.Chi-Square GOF 3.Chi-Square test of independence 4.Test for equality of means 5.Other

What test would be used to assess whether age is associated with Glucose Level? 1.ANOVA 2.Chi-Square GOF 3.Chi-Square test of independence 4.Test for equality of means 5.Other

In Framingham Heart Study, we want to assess risk factors for Diabetes  Consider a Tertiary Outcome = Diabetes Vs No Diabetes  Risk Factors Sex Age BMI (normal weight, overweight, obese) Genetics

What test would be used to assess whether sex is associated with Diabetes? 1.ANOVA 2.Chi-Square GOF 3.Chi-Square test of independence 4.Test for equality of means 5.Other

What test would be used to assess whether BMI is associated with Diabetes? 1.ANOVA 2.Chi-Square GOF 3.Chi-Square test of independence 4.Test for equality of means 5.Other

What test would be used to assess whether age is associated with Diabetes? 1.ANOVA 2.Chi-Square GOF 3.Chi-Square test of independence 4.Test for equality of means 5.Other

Correlation  Correlation (r)– measures the nature and strength of linear association between two variables at a time  Regression – equation that best describes relationship between variables

Simple Linear Regression  Y = Dependent, Outcome variable  X = Independent, Predictor variable  = b 0 + b 1 x  b 0 is the Y-intercept, b 1 is the slope

Simple Linear Regression Assumptions  Linear relationship between X and Y  Independence of errors  Homoscedasticity (constant variance) of the errors  Normality of errors

Multiple Linear Regression  Useful when we want to jointly examine the effect of several X variables on the outcome Y variable.  Y = continuous outcome variable  X 1, X 2, …, X p = set of independent or predictor variables .

Multiple Regression Analysis  Model is conditional, parameter estimates are conditioned on other variables in model  Perform overall test of regression If significant, examine individual predictors Relative importance of predictors by p- values (or standardized coefficients)

Multiple Regression Analysis  Predictors can be continuous, indicator variables (0/1) or a set of dummy variables  Dummy variables (for categorical predictors) Race: white, black, Hispanic  Black (1 if black, 0 otherwise)  Hispanic (1 if Hispanic, 0 otherwise)

Definitions  Confounding – the distortion of the effect of a risk factor on an outcome  Effect Modification – a different relationship between the risk factor and an outcome depending on the level of another variable

Multiple Regression for SBP: Comparison of Parameter Estimates Simple ModelsMultiple Regression  pp Age1.03 < <.0001 Male BMI1.80 < <.0001 BP Meds < <.0001 Focus on the association between BP meds and SBP…

RCT of New Drug to Raise HDL Example of Effect Modification WomenNMeanStd Dev New drug Placebo MenNMeanStd Dev New drug Placebo

Simple Logistic Regression  Outcome is dichotomous (binary)  We model the probability p of having the disease.

Multiple Logistic Regression  Outcome is dichotomous (1=event, 0=non-event) and p=P(event)  Outcome is modeled as log odds

Multiple Logistic Regression for Birth Defect (Y/N) Predictor bp OR (95% CI for OR) Intercept Smoke (0.34, 22.51) Age (1.02, 1.78) Interpretation of OR for age: The odds of having a birth defect for the older of two mothers differing in age by one year is estimated to be 1.35 times higher after adjusting for smoking.

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 longer survival.

Survival Analysis  Incomplete follow-up information  Censoring Measure follow-up time and not time to event We know survival time > follow-up time  Log rank test to compare survival in two or more independent groups

Survival Curve – Survival Function

Comparing Survival Curves H0: Two survival curves are equal 2 Test with df=1. Reject H 0 if 2 > 3.84 2 = Reject H 0.

Cox Proportional Hazards Model  Model: ln( h(t)/h 0 (t)) = b 1 X 1 + b 2 X 2 + … + b p X p  Exp(b i ) = hazard ratio  Model used to jointly assess effects of independent variables on outcome (time to an event).

NBPHE Questions?? Good Luck!