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Introduction to Biostatistics
i
Table of Contents
Introduction
1.
Why Biostatistics …………………………………………………………….………….
3
2.
Course Overview … …………………………………………………………………….. 9
Topic 1 - Summarizing Data
1. Scales of Measurement …………………………………………….……………………… 2
2. Descriptives for Nominal and Ordinal Data ………………………………………….…… 7
a. The Bar Chart …………………………………………….……………………….. 9
b. The Pie Chart ……………………………………………….…………………….. 13
3. Descriptives for Continuous Data ………………………………………..………….…….
a. The Histogram ……………………………………………………………..………..
b. The Frequency Polygon …………………………………………………………
c. The Cumulative Frequency Polygon ……………………………………………….
d. Percentiles (Quantiles) ………………………………………………………….
e. Quantile Quantile Plot ………………………………………………………………
f. Stem and Leaf Diagram ……………………………………………………….…
g. Box and Whisker Plot …………………………………………………………...
14
15
19
20
21
22
23
24
4. The Summation Notation…………………………………………………………………… 25
5. Measures of Central Tendency ………………………………………….………………
a. The mode ………………………………………………………………………..
b. The mean ……………………………………………………………….………..
c. The median ……………………………………………………………….……...
26
29
30
33
6. Measures of Dispersion ……………………………………………………………………
a. Variance ……………………………………………………………………………
b. Standard Deviation ………………………………………………………………..
c. Median Absolute Deviation from Median ………………………………………...
d. Standard Deviation v Standard Error ………………………………… ………….
e. A Feel for Sampling Distributions ……………………………………………….
f. The Coefficient of Variation ……………………………………………………..
g. The Range ………………………………………………………………………..
35
36
37
39
40
42
45
46
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Introduction to Biostatistics
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Topic 2 - Introduction to Probability
1. Why We Need Probability ………….……………………………………………………
2
2. Definition Probability Model ……………..………………………..……………………
4
3. The “Equally Likely” Setting: Introduction to
Probability Calculations ……..……………………..…………………………………….. 9
a. Sample space, Elementary Outcomes, Events…………………………………....……. 13
b. Types of Events ……………………………..………………………….……………… 16
4. Conditional Probability ……..……………………………………………………………... 19
a. Theorem of Total Probability …………………………………………………………... 23
b. Bayes Rule ……………………………………………………………………………… 25
5. A Simple Probability Model: The Bernoulli Distribution ………………………..……… 27
6. Probability in Diagnostic Testing ………………………………………….…..…………
a. Prevalence………………………………………………………….……………..
b. Incidence ………………………………….………………………………...….….
c. Sensitivity, Specificity ………………………………………….…………………
d. Predictive Value Positive, Negative Test …………………………….……………
29
29
29
30
33
7. Probability and Measures of Association for the 2x2 Table…………………………..….... 35
h. Risk ………………………………………………………………………….…….. 35
i. Odds ………………………………..………………………..…………………… 37
j. Relative Risk ……………………………………………………..………….…… 37
k. Odds Ratio ………………………………………………………………………...… 41
Appendices
1. Some Elementary Laws of Probability …………………………………….……….…… 44
2. Introduction to the Concept of Expected Value .…………………………….….………… 47
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Introduction to Biostatistics
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Topic 3 Populations and Samples
1. A Feeling for Populations v Samples ……………………………………………….……
2
2. Target Populations, Sampled Populations, Sampling Frames ……………………………
5
6. On Making Inferences from a Sample ……………………………………………………. 8
7. Simple Random Sampling ………………………………………………….……………. 10
8. Some Non-Probability Sampling Plans ………………………………………………….. 13
9. Introduction to Probability Sampling ……..………………………………………….….. 16
a. Sampling WITH v WITHOUT replacement ………………………………….… 17
b. How to select a simple random sample …………………………………….….. 25
10. Some Other Probability Sampling Plans …………………………………………….…… 28
a. Systematic ……………………………………………………………………….. 28
b. Stratified ………………………………………………………………….………. 30
C. Multi-stage ………………………………………………………….……………. 32
11. The Nationwide Inpatient Survey (NIS) …………………………………………..…….. 33
Topic 4 - The Bernoulli and Binomial Distributions
1. Review – What is a Discrete Probability Distribution…………………………………....
2
2. Statistical Expectation ………………………………………………………..…..……..
4
3. The Population Variance is a Statistical Expectation ………………………………….… 7
4. The Bernoulli Distribution …………………………………………………….………… 8
5. Introduction to Factorials and Combinatorials …………………………………….…… 11
6. The Binomial Distribution …………………………………………………..…………. 14
7. Illustration of the Binomial Distribution ….……………………………………………. 17
8. Resources for the Binomial Distribution ……………………………………………...
19
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Introduction to Biostatistics
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Topic 5 - The Normal Distribution
1. Introduction ……………………..……………………………………………..…………
3
2. Definition of the Normal Distribution ……………………………………….……..…..
4
3. Sample Average is Often Normally Distributed Introduction to the Central Limit Theorem 7
4. A Feel for the Normal Distribution ………………………………………….…………..
10
5. The Relevance of the Normal Distribution ………………………………………….….
12
6. Calculation of Probabilities for the Normal(0,1) ………………………………………
13
7. From Normal( µ , σ 2 ) to Normal(0,1) – The Z-Score …………………………….……
19
8. From Normal(0,1) to Normal( µ , σ 2 ) …………………………………………………… 22
Topic 6 - Estimation
1. Introduction ……………………..……………………………………..…..…..………… 2
2. Goals of Estimation ………………………………………...………………………..…..
5
3. Some notation/definitions ……………,………………………….……..……………….
7
4. How to Interpret a Confidence Interval …………………………….………………..…. 10
5. Normal: Confidence Interval for µ, σ2 Known ……………………………….………… 17
6. Introduction to the Student’s t-Distribution ……………………………………….……. 22
7. Normal: Confidence Interval for µ, σ2 Unknown …………………………….………… 28
8. Introduction to the Chi Square Distribution ……………………………………….……. 30
9. Normal: Confidence Interval for σ2 …………………………………………..…………. 36
continued
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Introduction to Biostatistics
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Topic 6 – Estimation continued
10. Normal Confidence Interval for µDIFFERENCE – Paired Data Setting …………………….…39
11. Normal: Confidence Interval for [µ1 - µ2] – Two Independent Groups …………………..44
12. Introduction to the F-Distribution ………………………………………….……………. 53
13. Normal: Confidence Interval for σ 12 σ 22 ……….…………………………….………….. 56
14. Binomial: Estimation of a Proportion π …………………………………….……………. 59
15. Binomial: Confidence Interval for [ π 1 − π 2 ] – Two Ind. Groups ……………………..…. 63
Appendix 1– Derivation of Confidence Interval for µ – Single Normal with
σ2 Known …………………………………………………………………………..
66
Appendix 2– Derivation of Confidence Interval for σ2 – Single Normal ……………………. 69
Appendix 3 – SE of a Binomial Proportion ……………………………………………………. 71
Topic 7 - Hypothesis Testing
1. The Logic of Hypothesis Testing ……………………………………..……………….
2
2. Beware the Statistical Hypothesis Test ………………………………………..………
15
3. Introduction to Type I, II Error and Statistical Power …………………….………….
18
4. Normal: Test for µ, σ2 Known …………………………………………………….…… 24
5. Normal: Test for µ, σ2 Unknown ………………………………………….…………… 27
6. Normal: Test for σ2 ………………………………………..……………………….…. 30
7. Normal Test for µDIFFERENCE – Paired Data Setting …………………………….…….… 32
8. Normal: Test for [µ1 - µ2] – Two Independent Groups ………………………….……… 36
continued
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Introduction to Biostatistics
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Topic 7 - Hypothesis Testing continued
9.
Normal: Test for Equality of Two Variances ( σ 12 σ 22 )…………..……………….…..
42
10. Single Binomial: Test for Proportion π …………………………………………..……. 45
11. Two Binomials:
Test for [ π 1 − π 2 ] – Two Ind. Groups ……………………………… 47
Appendix
URL’s for the Computation of Probabilities …………………………………………..……… 51
Topic 8 - Chi Square Tests
1. Introduction to Contingency Tables ……………………………………………………
2
2. Relationship Between the Normal(0,1) and the Chi Square Distribution ………………
9
3. Hypotheses of Independence or No Association ………………………………….…
4.
14
The Chi Square Test of No Association in an RxC Table ………………………….…..... 15
Topic 9 - Regression and Correlation
1. Definition of the Linear Regression Model ………………………………………….…..
2
2. Estimation ………………………………………………….……………………….….
10
3. The Analysis of Variance Table …………………………………………………….….
20
4. Assumptions for the Straight Line Regression ………………………………………..….
24
5. Hypothesis Testing ………………………………………………………………..……..
27
6. Confidence Interval Estimation ………………………………………………………....
35
7. Introduction to Correlation ……………………………………………………………….
39
8. Hypothesis Test for Correlation ……………………………………………………..….
42