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Examples Hypothesis Testing Exercise: Suppose the resting pulse rates for young women are normally distributed with mean μ = 66 and standard deviation σ = 9.2 beats per minute. A drug for the treatment of a medical condition is administered to 100 young women and their average pulse rate is found to be x = 68 beats per minute. Because the drug had for a long time been observed to increase pulse rates, test the claim that the drug does in fact increase the pulse rates. (i.e. H A is data based.) Solution: Step (1) H 0 : μ = 66 (the null hypothesis) Step (2) H A : μ > 66 (the research hypothesis) Step(3) x = 68 from sample data. Assuming H 0 is true, and noting that population standard deviation is known, standardising x leads to z = observed sample mean - null mean standard error of mean 219 Section 6
x − μ = σ / n 68 − 66 = 9.2 / 100 = 2.174 Step(4): Calculate p-value assuming μ = 66 0.50 66 68 X 0 2.174 Z p-value = Pr( X > 68 given μ = 66) 68 − 66 = Pr( Z > ) 9.2 / 100 = Pr( Z > 2.174) = 0. 015 This means that if H 0 is true, there is only a probability of 0.015 of observing a sample mean as large or larger than 68. Hence there is little support for H 0 . Reject H 0 and conclude the mean pulse rate has been increased by the treatment. 220 Section 6
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CONTENTS Introduction, General Info
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SECTION 9 This section introduces t
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Support Classes There is also a Wed
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Email Contact with Students From ti
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What is the best strategy for treat
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Examples • We use the frequency w
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In some experiments the investigato
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These studies are most often descri
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• Human studies: - anatomy and ph
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Biostatistics and research: an over
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Goal of health sciences professions
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Research A process for providing an
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Does early childhood circumcision r
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Steps in the research process: Deve
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Data Analysis and Computer Software
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1. Descriptive studies Aim: to desc
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Sample average = true mean + error
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Example: Options for studying the r
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Example RCT: LIPID study (NEJM, 199
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Example of cohort study: Study to i
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Example of case-control study Study
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3. Summary Classification of resear
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Types of data and graphical summari
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In these examples the data are said
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3. Rates, Ratios and Proportions Th
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Two treatments may be assessed from
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Example for Continuous Data: In a h
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Histograms: These are simple pictur
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(5) The relative (or percentage) fr
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Note The mean need not be one of th
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0 Median Mean The mean may be unsui
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If data are highly variable there a
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Example: Find the standard deviatio
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Box-and-whisker plot This is a seco
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Example: Thirty-two traps were plac
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Interpreting Box whisker plots (Ref
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How to make the call. This is based
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[C] Measures of Disease Frequency A
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Example: In a survey of eye disease
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2.1 Cumulative incidence is the pro
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• = Initiation of follow-up × =
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Example In (a)-(c) calculate a rele
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Relationship between prevalence and
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HIV/AIDS Many with HIV will live fo
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[D] Measures of disease association
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1. Relative effect = Relative Risk
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Example: A randomised trial of the
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Example: Data from a cohort study o
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Note Relative risks • provide inf
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SECTION 3 This section covers a bri
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A probability is therefore like a r
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An event is “more chemotherapy pa
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Pr(O) × Pr(O) × Pr(O) = 0.47 × 0
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• Event A and B denoted by A∩ B
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Independence The idea behind the te
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A tree diagram is useful for helpin
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The tree diagram shows all possible
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In probability notation we get: A i
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Pr(A) = 0.0001 (disease prevalence)
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1. Pr(B|A) is called the sensitivit
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Notice how the probability of heart
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WHO SARS Confirmed Result Yes No To
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“Have you ever had a sexually tra
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Probability Distribution and Random
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Describing Probability Distribution
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Example: A person infected with a d
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Note: Properties 3 and 4 tell us th
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Note: These results can be extended
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SECTION 4 This section introduces b
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Now suppose that we take a sample o
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Notes 1. If these conditions are me
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(a) Find the probability distributi
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or the teacher is at fault and more
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Solution (a) X is binomial with n =
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Example (Revision) The standard dru
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The Normal Distribution This distri
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μ Y Y = f(X) X The equation of suc
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Normal distribution calculations. T
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4. Pr(-1 < Z < 1.64) = Pr(0 < Z < 1
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Some calculations 1. Pr( μ X - σ
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(c) Find the diastolic blood pressu
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∴ Pr( X = x) ⎛ 1 1 ( x − 2)
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More on the normal and Statistical
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Example: It is claimed cancer tumor
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Transforming Data If data being ana
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Now suppose we want the range of va
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REVIEW EXERCISES 4. For the standar
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SECTION 5 This section defines samp
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Normal Underlying Population Contin
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The Distribution of Sample Means Su
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Distributions of means are for samp
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The addition rule for the variance
Page 186 and 187: Example: Suppose a population has v
Page 188 and 189: A Confidence Interval for the Mean
Page 190 and 191: Therefore, we are 95% confident tha
Page 192 and 193: Example: A pharmacologist is invest
Page 194 and 195: Use of t-table when σ X is unknown
Page 196 and 197: Student’s t distribution ν -t ν
Page 198 and 199: Example: Tablets must be produced w
Page 200 and 201: 100 Confidence Intervals (95%) Samp
Page 202 and 203: 185 Section 5
Page 206 and 207: Example: It is claimed that males c
Page 208 and 209: Case 2: Comparing means when sample
Page 210 and 211: Example 3: Following data are 24 ho
Page 212 and 213: Note: This confidence interval tell
Page 214 and 215: Find a 95% confidence interval for
Page 216 and 217: Example: Weight loss programme agai
Page 218 and 219: Confidence Intervals for a Proporti
Page 220 and 221: small sample, the normal distributi
Page 222 and 223: 0.5(1 − 0.5) 1.96 < 0.03 n 2 (1.9
Page 224 and 225: Solution: The proportions improved
Page 226 and 227: SOLUTIONS 2. (a) X is a normal dist
Page 228 and 229: [A] DISTRIBUTION SUMMARY 1. Binomia
Page 230 and 231: [B] SUMMARY: CONFIDENCE INTERVALS s
Page 232 and 233: SECTION 6 This section reviews hypo
Page 234 and 235: If the data suggest harm we are lik
Page 238 and 239: Notes: 1. R-cmdr and other statisti
Page 240 and 241: Step(3): Sample gives p = 27/293 =
Page 242 and 243: Step(3): Sample gives x − x = 3.4
Page 246 and 247: Notes on Hypothesis Testing 1. Ther
Page 248 and 249: (b) H 0 is not true, and the observ
Page 250 and 251: (b) Mean Diff = 36. Confidence inte
Page 252 and 253: Summary of previous results 0 = nul
Page 254 and 255: Example A clinical trial is set up
Page 256 and 257: Errors in Hypothesis Testing The le
Page 258 and 259: Some Computer packages (Minitab is
Page 260 and 261: A printout is as follows: Power and
Page 262 and 263: (c) Halve the value of sigma to 1.0
Page 264 and 265: Revision Examples 1. Exam 2006: In
Page 266 and 267: (c) (d) (e) (2 marks) A mean temper
Page 268 and 269: 2. Exam 2005 An ecologist must dete
Page 270 and 271: (c) (2 marks) Find the probability
Page 272 and 273: SECTION 7 One factor analysis of va
Page 274 and 275: Example A general surgeon believes
Page 276 and 277: We now investigate the problem of h
Page 278 and 279: Estimation of Components (for refer
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Page 282 and 283: υ 2 υ 1 1 2 3 4 … 30 1 15 3.05
Page 284 and 285: If n 1 = n 2 = … = n k = n (say)
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[ 2 2 2 2 2 ] s + s + s + s s 2 1 s
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Total SS = 4 2 + 12 2 + … + 89 2
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POST ANALYSIS OF VARIANCE RESULTS I
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Solution: x = 110/4 = 27.5 x = 285/
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The next two residual plots confirm
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SECTION 8 This section covers the a
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Examples of research questions that
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Example 1: What is the prevalence o
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Example 3: Is there an association
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Prospective Studies • groups are
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Confidence interval for relative ri
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Confidence interval for attributabl
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Case control studies • a group of
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a/ b The odds ratio, OR = c/ d = 32
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so a a c c ≈ and ≈ b a+ b d c+
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Chi Square Test for Contingency Tab
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We wish to know whether the data in
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Using this formula, expected number
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Example: For the drug responses, χ
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• the p-value gives the probabili
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Example 5 Is there an association b
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χ 2 2 (100−108.99) (107−111.01
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This page for reference only Income
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(a) OR = 1.90 with CI (1.23, 2.92)
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0 1 1.5 2 3 3.5 a x b x c d e x x x
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with υ = 1 degree of freedom. Sinc
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COMBINED ADMIT DECLINE Male 490 (44
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SOLUTIONS 29 1. (b) Risk (aspirin g
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SECTION 9 This section introduces t
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Regression methods are used to intr
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BAC X X X • • • • X X X X X
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FEV + + + + + + + + + + + + + + + +
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80 Y X 70 60 X X X 50 X X 40 X 100
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The method of least squares finds t
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N.B. 1. In this situation we have r
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important as they represent the err
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To illustrate, the example has x =
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That is, the total variation is par
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The F-distribution (Table in Append
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5. For the validity of the F-test,
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2. The divisor is (n - 2) rather th
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x i y i ( y − yˆ ) i i ( y − 1
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Stress (X) 55 94 64 73 96 86 Blood
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Confidence Interval for Prediction
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Y Prediction Interval X EXAMPLE: 20
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(c) (3 marks) What is the predicted
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(b) Estimated standard error = 3.47
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The denominator in formula for r is
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2 ( y i − y) ( x − x)( y − y)
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2. r = -1 is smallest value which i
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Some examples on correlation and as
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REVIEW EXERCISES 1. Physical fitnes
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SECTION 10 Multiple regression mode
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• The type of multiple regression
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The multiple linear regression mode
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Example For lung transplantation it
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It appears total lung capacity incr
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Step 3: Fit (in R-cmdr) Multiple Li
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The fitted equation is therefore ŷ
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It is 6.98 - 5.20 = 1.78 litres whe
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Are all three variables needed in t
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Test of the hypothesis H 0 : β 1 =
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95% confidence interval for coeffic
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Checking the fit of the model We do
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The points in the normal P-P plot l
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Analysis of Covariance This analysi
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At this stage the ages have been ig
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[C] Third, a regression analysis of
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The confidence interval here is the
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Notes: 1. 2 ˆβ (the coefficient o
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The logistic regression model is:
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(1) Investigating the relationship
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BUT what about the potential confou
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SECTION 11 Study design principles,
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Critical appraisal Critical apprais
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1. Introduction to critical apprais
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2. Design and appraisal of Descript
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Recall Suppose we want to estimate
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Selection bias • systematic error
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Example Life in New Zealand Survey,
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Internal validity Bias Selection bi
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Study design and critical appraisal
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Types of design: • experimental (
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Example Does smoking cause coronary
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Assessing internal validity: Bias S
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Evaluation and control of bias •
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Example of confounding: A study was
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Positive and Negative Confounding P
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Some comments on confounding: AGE a
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(3) Control of confounding during t
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Basic structure of a RCT • popula
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The way in which patients are alloc
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Randomised controlled trials : Exam
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Key results 849 randomised placebo
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Internal validity Chance • 95% co
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Intention-to-treat analysis “once
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Study design and critical appraisal
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Prospective cohort study (concurren
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Selection of participants • 5,766
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Bias Selection bias • because the
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Case-control studies Ref: “Case-C
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Selection of participants Study pop
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Key results Response rates: Cases:
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In this study: • the odds ratios
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Information bias Things done to min
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Generalisability Could we apply the
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Appendix One: The Basics This appen
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Example 3 If Z = X − μ calculate
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There are no hard and fast rules co
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Converting Fractions to Decimals Th
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X − μ Example 1: If Z = , calcul
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7. Sigma means Add Up The Greek let
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Section 2: Basic Statistical Concep
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5. Quartiles and Interquartile Rang
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9. If 1.96 X − 2.8 = , calculate
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• Probability of B given A: Pr( A
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Summary of Formulae 1. Normal Distr
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