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

2. 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An archived version of this review, along with the PPT file, will be available on the NBPHE website ( under Study Resources

3. 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.

4. 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

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

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

7. Numerical Summaries of Dichotomous, Categorical and Ordinal Variables
Frequency Distribution Table
Heath Status
Freq.
Rel. Freq.
Cumulative Freq
Cumulative Rel. Freq.
Excellent 19
38%
Very Good 12
24% 31
62%
Good 9
18% 40
80%
Fair 6
12% 46
92%
Poor 4 8% 50
100%
n=50
Ordinal variables only

8. Frequency Bar Chart

9. Relative Frequency Histogram

10. Continuous Variables
Assume, in theory, any value between a theoretical minimum and maximum
Quantitative, measurement variables
Example – systolic blood pressure
Standard Summary: n = 75, X = 123.6, s = 19.4
Second sample n = 75, X = 128.1, s = 6.4

11. 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 = Q3-Q1
Outliers <Q1–1.5 IQR or >Q3+1.5 IQR

12. Mean Vs. Median

13. Box and Whisker Plot
Min Q Median Q Max

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

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

16. The distribution of SBP in men, 20-29 years is shown below
The distribution of SBP in men, years is shown below. What is the best summary of a typical value
Mean
Median
Interquartile range
Standard Deviation

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

18. The best summary of variability for the following continuous variable is
Mean
Median
Interquartile range
Standard Deviation

19. Numerical and Graphical Summaries
Dichotomous and categorical
Frequencies and relative frequencies
Bar charts (freq. or relative freq.)
Ordinal
Frequencies, relative frequencies, cumulative frequencies and cumulative relative frequencies
Histograms (freq. or relative freq.
Continuous
n, X and s or median and IQR (if outliers)
Box whisker plot

20. What is the probability of selecting a male with optimal blood pressure?
Blood Pressure Category
Optimal Normal Pre-Htn Htn Total
Male
Female
Total
20/25
20/80
20/150

21. What is the probability of selecting a patient with Pre-Htn or Htn?
Blood Pressure Category
Optimal Normal Pre-Htn Htn Total
Male
Female
Total
95/150
45/80
55/150

22. What proportion of men have prevalent CVD?
CVD Free of CVD
Men
Women
35/80
35/265
35/300

23. What proportion of patients with CVD are men ?
CVD Free of CVD
Men
Women
35/700
35/80
80/300

24. 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 History No
Yes
215 25 90 15
P(Current CVD| Family Hx) = 15/105 = 0.143
P(Current CVD| No Family Hx) = 25/240 = 0.104

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

26. Probability Models – Binomial Distribution
Two possible outcomes: success and failure
Replications of process are independent
P(success) is constant for each replication
Mean=np, variance=np(1-p)

27. Probability Models – Poisson Distribution
Two possible outcomes: success and failure
Replications of process are independent
Often used to model counts (often used to model rare events)
Mean=m, variance=m

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

29. Normal Distribution Properties of Normal Distribution
I) The normal distribution is symmetric about the mean (i.e., P(X > m) = P(X < m) = 0.5).
ii) The mean and variance (m and s2) 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

30. 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

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

32. Standard Normal Distribution Z
Normal distribution with m=0 and s=1

33. Normal Distribution P(X<30)= P(Z<0.17) = 0.5675
From a table of standard normal probabilities or statistical computing package.

34. 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 ?

35. Normal Distribution ZM = (140 - 108) / 14 = 2.29
ZF = ( ) / 8 = 2.50
Which is more extreme?

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

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

38. Central Limit Theorem (Non-normal) population with m, s
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

39. 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.

40. 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

41. 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)

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

43. P-values P-values represent the exact significance of the data
Estimate p-values when rejecting H0 to summarize significance of the data (approximate with statistical tables, exact value with computing package)
If p < a then reject H0

44. Errors in Hypothesis Tests
Conclusion of Statistical Test
Do Not Reject H0 Reject H0
H0 true Correct Type I error
H0 false Type II error Correct

45. Continuous Outcome Confidence Interval for m
Continuous outcome - 1 Sample
n > 30
n < 30
Example.
95% CI for mean waiting time at ED
Data: n=100, X =37.85 and s=9.5 mins
(35.99 to 39.71)
Statistical computing packages use t throughout.

46. 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,

47. 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)

48. One Sample Procedures – Comparisons with Historical/External Control
Continuous Dichotomous
H0: m=m0 H0: p=p0
H1: m>m0, <m0, ≠m0 H1: p>p0, <p0, ≠p0
n>30
n<30

49. One Sample Procedures – Comparisons with Historical/External Control
Categorical or Ordinal outcome
c2 Goodness of fit test
H0: p1=p10, p2=p20, , pk=pk0
H1: H0 is false

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

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

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

53. Continuous Outcome Confidence Interval for (m1-m2)
Continuous outcome - 2 Independent Samples
n1>30 and n2>30
n1<30 or n2<30

54. Hypothesis Testing for (m1-m2)
Continuous outcome
2 Independent Sample
H0: m1=m2 (m1-m2 = 0)
H1: m1>m2, m1<m2, m1≠m2

55. Hypothesis Testing for (m1-m2)
Test Statistic
n1>30 and n2> 30
n1<30 or n2<30

56. An RCT is planned to show the efficacy of a new drug vs
An RCT is planned to show the efficacy of a new drug vs. placebo to lower total cholesterol.
What are the hypotheses?
H0: mP=mN H1: mP>mN
H0: mP=mN H1: mP<mN
H0: mP=mN H1: mP≠mN

57. 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)

58. Dichotomous Outcome Confidence Interval for (p1-p2)
Dichotomous outcome - 2 Independent Samples

59. 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)

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

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

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

63. Hypothesis Testing for (p1-p2)
Dichotomous outcome
2 Independent Sample
H0: p1=p2
H1: p1>p2, p1<p2, p1≠p2
Test Statistic

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

65. 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=X1-X2)

66. Two Dependent/Matched Samples
Subject ID Measure 1 Measure 2 .
Measures taken serially in time or under different experimental conditions

67. Crossover Trial Treatment Treatment Eligible R Participants
Placebo Placebo
Each participant measured on Treatment and placebo

68. Confidence Intervals for md
Continuous outcome
2 Matched/Paired Samples
n > 30
n < 30

69. Hypothesis Testing for md
Continuous outcome
2 Matched/Paired Samples
H0: md=0
H1: md>0, md<0, md≠0
Test Statistic
n>30
n<30

70. Independent Vs Matched Design

71. 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)

72. The null value of a difference in means is…
0.5 1 2

73. The null value of a mean difference is…
0.5 1 2

74. The null value of a relative risk is…
0.5 1 2

75. The null value of a difference in proportions is…
0.5 1 2

76. The null value of an odds ratio is…
0.5 1 2

77. A two sided test for the equality of means produces p=0.20. Reject H0?
Yes No
Maybe

78. Hypothesis Testing for More than 2 Means - Analysis of Variance
Continuous outcome
k Independent Samples, k > 2
H0: m1=m2=m3 … =mk
H1: Means are not all equal
Test Statistic
F is ratio of between group variation to within group variation (error)

79. ANOVA Table Source of Sums of Mean Variation Squares df Squares F
Between
Treatments k-1 SSB/k-1 MSB/MSE
Error N-k SSE/N-k
Total N-1

80. 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

81. 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

82. 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)

83. c2 Test of Independence Dichotomous, ordinal or categorical outcome
2 or More Samples
H0: The distribution of the outcome is independent of the groups
H1: H0 is false
Test Statistic

84. c2 Test of Independence Data organization (r by c table)
Is there distribution of the outcome different (associated with) groups
Outcome
Group 1 2 3 A
20%
40% B
50%
25% C
90% 5%

85. What Tests Were Used?

86. 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

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

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

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

90. Consider a Secondary Outcome = Fasting Glucose Level Risk Factors
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

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

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

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

94. 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

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

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

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

98. 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

99. What is the most likely value of r for the data shown below?

100. What is the most likely value of r for the data shown below?

101. Simple Linear Regression
Y = Dependent, Outcome variable
X = Independent, Predictor variable
= b0 + b1 x
b0 is the Y-intercept, b1 is the slope

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

103. 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
X1, X2, …, Xp = set of independent or predictor variables .

104. 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)

105. 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)

106. 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

107. Multiple Regression for SBP: Comparison of Parameter Estimates
Simple Models Multiple Regression
b p b p
Age < <.0001
Male
BMI < <.0001
BP Meds < <.0001
Focus on the association between BP meds and SBP…

108. RCT of New Drug to Raise HDL Example of Effect Modification
Women N
Mean
Std Dev
New drug 40
38.88
3.97
Placebo 41
39.24
4.21
Men 10
45.25
1.89 9
39.06
2.22

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

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

111. Multiple Logistic Regression for Birth Defect (Y/N)
Predictor b p 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.

112. 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.

113. 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

114. Survival Curve – Survival Function

115. Comparing Survival Curves
H0: Two survival curves are equal
c2 Test with df=1. Reject H0 if c2 > 3.84
c2 = Reject H0.

116. Cox Proportional Hazards 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).

117. Outcome= all-cause mortality
Age and Sex as predictors
bi p HR
Age
Male Sex

118. Sample Size Determination
Need sample to ensure precision in analysis
Sample size determined based on type of planned analysis CI
Test of hypothesis

119. Determining Sample Size for Confidence Interval Estimates
Goal is to estimate an unknown parameter using a confidence interval estimate
Plan a study to sample individuals, collect appropriate data and generate CI estimate
How many individuals should we sample?

120. Determining Sample Size for Confidence Interval Estimates
Confidence intervals:
point estimate + margin of error
Determine n to ensure small margin of error (precision) – accounting for attrition!
Must specify desired margin of error, confidence level and variability of parameter

121. Determining Sample Size for Hypothesis Testing
How many participants are needed to ensure that there is a high probability of rejecting H0 when it is really false?
Determine n to ensure high power (usually 80% or 90%) – accounting for attrition!
Must specify desired power, a and effect size (difference in parameter under H0 versus H1)

122. Determining Sample Size for Hypothesis Testing
b and Power are related to the sample size, level of significance (a) and the effect size (difference in parameter of interest under H0 versus H1)
Power is higher with larger a
Power is higher with larger effect size
Power is higher with larger sample size

123. Sample Size Determination
Critical
Ethical
Sometimes difficult