Medical Statistics from Scratch
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Medical Statistics from Scratch
An Introduction for Health Professionals
Bowers, David
John Wiley & Sons Inc
10/2019
496
Mole
Inglês
9781119523888
15 a 20 dias
990
Descrição não disponível.
Preface to the 4th Edition xix
Preface to the 3rd Edition xxi
Preface to the 2nd Edition xxiii
Preface to the 1st Edition xxv
Introduction xxvii
I Some Fundamental Stuff 1
1 First things first - the nature of data 3
Variables and data 3
Where are we going ...? 5
The good, the bad, and the ugly - types of variables 5
Categorical data 6
Nominal categorical data 6
Ordinal categorical data 7
Metric data 8
Discrete metric data 8
Continuous metric data 9
How can I tell what type of variable I am dealing with? 10
The baseline table 11
II Descriptive Statistics 15
2 Describing data with tables 17
Descriptive statistics. What can we do with raw data? 18
Frequency tables - nominal data 18
The frequency distribution 19
Relative frequency 20
Frequency tables - ordinal data 20
Frequency tables - metric data 22
Frequency tables with discrete metric data 22
Cumulative frequency 24
Frequency tables with continuous metric data - grouping the raw data 25
Open-ended groups 27
Cross-tabulation - contingency tables 28
Ranking data 30
3 Every picture tells a story - describing data with charts 31
Picture it! 32
Charting nominal and ordinal data 32
The pie chart 32
The simple bar chart 34
The clustered bar chart 35
The stacked bar chart 37
Charting discrete metric data 39
Charting continuous metric data 39
The histogram 39
The box (and whisker) plot 42
Charting cumulative data 44
The cumulative frequency curve with discrete metric data 44
The cumulative frequency curve with continuous metric data 44
Charting time-based data - the time series chart 47
The scatterplot 48
The bubbleplot 49
4 Describing data from its shape 51
The shape of things to come 51
Skewness and kurtosis as measures of shape 52
Kurtosis 55
Symmetric or mound-shaped distributions 56
Normalness - the Normal distribution 56
Bimodal distributions 58
Determining skew from a box plot 59
5 Measures of location - Numbers R us 62
Numbers, percentages, and proportions 62
Preamble 63
N umbers, percentages, and proportions 64
Handling percentages - for those of us who might need a reminder 65
Summary measures of location 67
The mode 68
The median 69
The mean 70
Percentiles 71
Calculating a percentile value 72
What is the most appropriate measure of location? 73
6 Measures of spread - Numbers R us - (again) 75
Preamble 76
The range 76
The interquartile range (IQR) 76
Estimating the median and interquartile range from the cumulative frequency curve 77
The boxplot (also known as the box and whisker plot) 79
Standard deviation 82
Standard deviation and the Normal distribution 84
Testing for Normality 86
Using SPSS 86
Using Minitab 87
Transforming data 88
7 Incidence, prevalence, and standardisation 92
Preamble 93
The incidence rate and the incidence rate ratio (IRR) 93
The incidence rate ratio 94
Prevalence 94
A couple of difficulties with measuring incidence and prevalence 97
Some other useful rates 97
Crude mortality rate 97
Case fatality rate 98
Crude maternal mortality rate 99
Crude birth rate 99
Attack rate 99
Age-specific mortality rate 99
Standardisation - the age-standardised mortality rate 101
The direct method 102
The standard population and the comparative mortality ratio (CMR) 103
The indirect method 106
The standardised mortality rate 107
III The Confounding Problem 111
8 Confounding - like the poor, (nearly) always with us 113
Preamble 114
What is confounding? 114
Confounding by indication 117
Residual confounding 119
Detecting confounding 119
Dealing with confounding - if confounding is such a problem, what can we do about it? 120
Using restriction 120
Using matching 121
Frequency matching 121
One-to-one matching 121
Using stratification 122
Using adjustment 122
Using randomisation 122
IV Design and Data 125
9 Research design - Part I: Observational study designs 127
Preamble 128
Hey ho! Hey ho! it's off to work we go 129
Types of study 129
Observational studies 130
Case reports 130
Case series studies 131
Cross-sectional studies 131
Descriptive cross-sectional studies 132
Confounding in descriptive cross-sectional studies 132
Analytic cross-sectional studies 133
Confounding in analytic cross-sectional studies 134
From here to eternity - cohort studies 135
Confounding in the cohort study design 139
Back to the future - case-control studies 139
Confounding in the case-control study design 141
Another example of a case-control study 142
Comparing cohort and case-control designs 143
Ecological studies 144
The ecological fallacy 145
10 Research design - Part II: getting stuck in - experimental studies 146
Clinical trials 147
Randomisation and the randomised controlled trial (RCT) 148
Block randomisation 149
Stratification 149
Blinding 149
The crossover RCT 150
Selection of participants for an RCT 153
Intention to treat analysis (ITT) 154
11 Getting the participants for your study: ways of sampling 156
From populations to samples - statistical inference 157
Collecting the data - types of sample 158
The simple random sample and its offspring 159
The systematic random sample 159
The stratified random sample 160
The cluster sample 160
Consecutive and convenience samples 161
How many participants should we have? Sample size 162
Inclusion and exclusion criteria 162
Getting the data 163
V Chance Would Be a Fine Thing 165
12 The idea of probability 167
Preamble 167
Calculating probability - proportional frequency 168
Two useful rules for simple probability 169
Rule 1. The multiplication rule for independent events 169
Rule 2. The addition rule for mutually exclusive events 170
Conditional and Bayesian statistics 171
Probability distributions 171
Discrete versus continuous probability distributions 172
The binomial probability distribution 172
The Poisson probability distribution 173
The Normal probability distribution 174
13 Risk and odds 175
Absolute risk and the absolute risk reduction (ARR) 176
The risk ratio 178
The reduction in the risk ratio (or relative risk reduction (RRR)) 178
A general formula for the risk ratio 179
Reference value 179
N umber needed to treat (NNT) 180
What happens if the initial risk is small? 181
Confounding with the risk ratio 182
Odds 183
Why you can't calculate risk in a case-control study 185
The link between probability and odds 186
The odds ratio 186
Confounding with the odds ratio 189
Approximating the risk ratio from the odds ratio 189
VI The Informed Guess - An Introduction to Confidence Intervals 191
14 Estimating the value of a single population parameter - the idea of confidence intervals 193
Confidence interval estimation for a population mean 194
The standard error of the mean 195
How we use the standard error of the mean to calculate a confidence interval for a population mean 197
Confidence interval for a population proportion 200
Estimating a confidence interval for the median of a single population 203
15 Using confidence intervals to compare two population parameters 206
What's the difference? 207
Comparing two independent population means 207
An example using birthweights 208
Assessing the evidence using the confidence interval 211
Comparing two paired population means 215
Within-subject and between-subject variations 215
Comparing two independent population proportions 217
Comparing two independent population medians - the Mann-Whitney rank sums method 219
Comparing two matched population medians - the Wilcoxon signed-ranks method 220
16 Confidence intervals for the ratio of two population parameters 224
Getting a confidence interval for the ratio of two independent population means 225
Confidence interval for a population risk ratio 226
Confidence intervals for a population odds ratio 229
Confidence intervals for hazard ratios 232
VII Putting it to the Test 235
17 Testing hypotheses about the difference between two population parameters 237
Answering the question 238
The hypothesis 238
The null hypothesis 239
The hypothesis testing process 240
The p-value and the decision rule 241
A brief summary of a few of the commonest tests 242
Using the p-value to compare the means of two independent populations 244
Interpreting computer hypothesis test results for the difference in two independent population means - the two-sample t test 245
Output from Minitab - two-sample t test of difference in mean birthweights of babies born to white mothers and to non-white mothers 245
Output from SPSS_: two-sample t test of difference in mean birthweights of babies born to white mothers and to non-white mothers 246
Comparing the means of two paired populations - the matched-pairs t test 248
Using p-values to compare the medians of two independent populations: the Mann-Whitney rank-sums test 248
How the Mann-Whitney test works 249
Correction for multiple comparisons 250
The Bonferroni correction for multiple testing 250
Interpreting computer output for the Mann-Whitney test 252
With Minitab 252
With SPSS 252
Two matched medians - the Wilcoxon signed-ranks test 254
Confidence intervals versus hypothesis testing 254
What could possibly go wrong? 255
Types of error 256
The power of a test 257
Maximising power - calculating sample size 258
Rule of thumb 1. Comparing the means of two independent populations (metric data) 258
Rule of thumb 2. Comparing the proportions of two independent populations (binary data) 259
18 The Chi-squared (?2) test - what, why, and how? 261
Of all the tests in all the world - you had to walk into my hypothesis testing procedure 262
Using chi-squared to test for related-ness or for the equality of proportions 262
Calculating the chi-squared statistic 265
Using the chi-squared statistic 267
Yate's correction (continuity correction) 268
Fisher's exact test 268
The chi-squared test with Minitab 269
The chi-squared test with SPSS 270
The chi-squared test for trend 272
SPSS output for chi-squared trend test 274
19 Testing hypotheses about the ratio of two population parameters 276
Preamble 276
The chi-squared test with the risk ratio 277
The chi-squared test with odds ratios 279
The chi-squared test with hazard ratios 281
VIII Becoming Acquainted 283
20 Measuring the association between two variables 285
Preamble - plotting data 286
Association 287
The scatterplot 287
The correlation coefficient 290
Pearson's correlation coefficient 290
Is the correlation coefficient statistically significant in the population? 292
Spearman's rank correlation coefficient 294
21 Measuring agreement 298
To agree or not agree: that is the question 298
Cohen's kappa (?) 300
Some shortcomings of kappa 303
Weighted kappa 303
Measuring the agreement between two metric continuous variables, the Bland-Altmann plot 303
IX Getting into a Relationship 307
22 Straight line models: linear regression 309
Health warning! 310
Relationship and association 310
A causal relationship - explaining variation 312
Refresher - finding the equation of a straight line from a graph 313
The linear regression model 314
First, is the relationship linear? 315
Estimating the regression parameters - the method of ordinary least squares (OLS) 316
Basic assumptions of the ordinary least squares procedure 317
Back to the example - is the relationship statistically significant? 318
Using SPSS to regress birthweight on mother's weight 318
Using Minitab 319
Interpreting the regression coefficients 320
Goodness-of-fit, R2 320
Multiple linear regression 322
Adjusted goodness-of-fit: R?2 324
Including nominal covariates in the regression model: design variables and coding 326
Building your model. Which variables to include? 327
Automated variable selection methods 328
Manual variable selection methods 329
Adjustment and confounding 330
Diagnostics - checking the basic assumptions of the multiple linear regression model 332
Analysis of variance 333
23 Curvy models: logistic regression 334
A second health warning! 335
The binary outcome variable 335
Finding an appropriate model when the outcome variable is binary 335
The logistic regression model 337
Estimating the parameter values 338
Interpreting the regression coefficients 338
Have we got a significant result? statistical inference in the logistic regression model 340
The Odds Ratio 341
The multiple logistic regression model 343
Building the model 344
Goodness-of-fit 346
24 Counting models: Poisson regression 349
Preamble 350
Poisson regression 350
The Poisson regression equation 351
Estimating ?1 and ?2 with the estimators b0 and b1 352
Interpreting the estimated coefficients of a Poisson regression, b0 and b1 352
Model building - variable selection 355
Goodness-of-fit 357
Zero-inflated Poisson regression 358
Negative binomial regression 359
Zero-inflated negative binomial regression 361
X Four More Chapters 363
25 Measuring survival 365
Preamble 366
Censored data 366
A simple example of survival in a single group 366
Calculating survival probabilities and the proportion surviving: the Kaplan-Meier table 368
The Kaplan-Meier curve 369
Determining median survival time 369
Comparing survival with two groups 370
The log-rank test 371
An example of the log-rank test in practice 372
The hazard ratio 372
The proportional hazards (Cox's) regression model - introduction 373
The proportional hazards (Cox's) regression model - the detail 376
Checking the assumptions of the proportional hazards model 377
An example of proportional hazards regression 377
26 Systematic review and meta-analysis 380
Introduction 381
Systematic review 381
The forest plot 383
Publication and other biases 384
The funnel plot 386
Significance tests for bias - Begg's and Egger's tests 387
Combining the studies: meta-analysis 389
The problem of heterogeneity - the Q and I2 tests 389
27 Diagnostic testing 393
Preamble 393
The measures - sensitivity and specificity 394
The positive prediction and negative prediction values (PPV and NPV) 395
The sensitivity-specificity trade-off 396
Using the ROC curve to find the optimal sensitivity versus specificity trade-off 397
28 Missing data 400
The missing data problem 400
Types of missing data 403
Missing completely at random (MCAR) 403
Missing at Random (MAR) 403
Missing not at random (MNAR) 404
Consequences of missing data 405
Dealing with missing data 405
Do nothing - the wing and prayer approach 406
List-wise deletion 406
Pair-wise deletion 407
Imputation methods - simple imputation 408
Replacement by the Mean 408
Last observation carried forward 409
Regression-based imputation 410
Multiple imputation 411
Full Information Maximum Likelihood (FIML) and other methods 412
Appendix: Table of random numbers 414
References 415
Solutions to Exercises 424
Index 457
Preface to the 3rd Edition xxi
Preface to the 2nd Edition xxiii
Preface to the 1st Edition xxv
Introduction xxvii
I Some Fundamental Stuff 1
1 First things first - the nature of data 3
Variables and data 3
Where are we going ...? 5
The good, the bad, and the ugly - types of variables 5
Categorical data 6
Nominal categorical data 6
Ordinal categorical data 7
Metric data 8
Discrete metric data 8
Continuous metric data 9
How can I tell what type of variable I am dealing with? 10
The baseline table 11
II Descriptive Statistics 15
2 Describing data with tables 17
Descriptive statistics. What can we do with raw data? 18
Frequency tables - nominal data 18
The frequency distribution 19
Relative frequency 20
Frequency tables - ordinal data 20
Frequency tables - metric data 22
Frequency tables with discrete metric data 22
Cumulative frequency 24
Frequency tables with continuous metric data - grouping the raw data 25
Open-ended groups 27
Cross-tabulation - contingency tables 28
Ranking data 30
3 Every picture tells a story - describing data with charts 31
Picture it! 32
Charting nominal and ordinal data 32
The pie chart 32
The simple bar chart 34
The clustered bar chart 35
The stacked bar chart 37
Charting discrete metric data 39
Charting continuous metric data 39
The histogram 39
The box (and whisker) plot 42
Charting cumulative data 44
The cumulative frequency curve with discrete metric data 44
The cumulative frequency curve with continuous metric data 44
Charting time-based data - the time series chart 47
The scatterplot 48
The bubbleplot 49
4 Describing data from its shape 51
The shape of things to come 51
Skewness and kurtosis as measures of shape 52
Kurtosis 55
Symmetric or mound-shaped distributions 56
Normalness - the Normal distribution 56
Bimodal distributions 58
Determining skew from a box plot 59
5 Measures of location - Numbers R us 62
Numbers, percentages, and proportions 62
Preamble 63
N umbers, percentages, and proportions 64
Handling percentages - for those of us who might need a reminder 65
Summary measures of location 67
The mode 68
The median 69
The mean 70
Percentiles 71
Calculating a percentile value 72
What is the most appropriate measure of location? 73
6 Measures of spread - Numbers R us - (again) 75
Preamble 76
The range 76
The interquartile range (IQR) 76
Estimating the median and interquartile range from the cumulative frequency curve 77
The boxplot (also known as the box and whisker plot) 79
Standard deviation 82
Standard deviation and the Normal distribution 84
Testing for Normality 86
Using SPSS 86
Using Minitab 87
Transforming data 88
7 Incidence, prevalence, and standardisation 92
Preamble 93
The incidence rate and the incidence rate ratio (IRR) 93
The incidence rate ratio 94
Prevalence 94
A couple of difficulties with measuring incidence and prevalence 97
Some other useful rates 97
Crude mortality rate 97
Case fatality rate 98
Crude maternal mortality rate 99
Crude birth rate 99
Attack rate 99
Age-specific mortality rate 99
Standardisation - the age-standardised mortality rate 101
The direct method 102
The standard population and the comparative mortality ratio (CMR) 103
The indirect method 106
The standardised mortality rate 107
III The Confounding Problem 111
8 Confounding - like the poor, (nearly) always with us 113
Preamble 114
What is confounding? 114
Confounding by indication 117
Residual confounding 119
Detecting confounding 119
Dealing with confounding - if confounding is such a problem, what can we do about it? 120
Using restriction 120
Using matching 121
Frequency matching 121
One-to-one matching 121
Using stratification 122
Using adjustment 122
Using randomisation 122
IV Design and Data 125
9 Research design - Part I: Observational study designs 127
Preamble 128
Hey ho! Hey ho! it's off to work we go 129
Types of study 129
Observational studies 130
Case reports 130
Case series studies 131
Cross-sectional studies 131
Descriptive cross-sectional studies 132
Confounding in descriptive cross-sectional studies 132
Analytic cross-sectional studies 133
Confounding in analytic cross-sectional studies 134
From here to eternity - cohort studies 135
Confounding in the cohort study design 139
Back to the future - case-control studies 139
Confounding in the case-control study design 141
Another example of a case-control study 142
Comparing cohort and case-control designs 143
Ecological studies 144
The ecological fallacy 145
10 Research design - Part II: getting stuck in - experimental studies 146
Clinical trials 147
Randomisation and the randomised controlled trial (RCT) 148
Block randomisation 149
Stratification 149
Blinding 149
The crossover RCT 150
Selection of participants for an RCT 153
Intention to treat analysis (ITT) 154
11 Getting the participants for your study: ways of sampling 156
From populations to samples - statistical inference 157
Collecting the data - types of sample 158
The simple random sample and its offspring 159
The systematic random sample 159
The stratified random sample 160
The cluster sample 160
Consecutive and convenience samples 161
How many participants should we have? Sample size 162
Inclusion and exclusion criteria 162
Getting the data 163
V Chance Would Be a Fine Thing 165
12 The idea of probability 167
Preamble 167
Calculating probability - proportional frequency 168
Two useful rules for simple probability 169
Rule 1. The multiplication rule for independent events 169
Rule 2. The addition rule for mutually exclusive events 170
Conditional and Bayesian statistics 171
Probability distributions 171
Discrete versus continuous probability distributions 172
The binomial probability distribution 172
The Poisson probability distribution 173
The Normal probability distribution 174
13 Risk and odds 175
Absolute risk and the absolute risk reduction (ARR) 176
The risk ratio 178
The reduction in the risk ratio (or relative risk reduction (RRR)) 178
A general formula for the risk ratio 179
Reference value 179
N umber needed to treat (NNT) 180
What happens if the initial risk is small? 181
Confounding with the risk ratio 182
Odds 183
Why you can't calculate risk in a case-control study 185
The link between probability and odds 186
The odds ratio 186
Confounding with the odds ratio 189
Approximating the risk ratio from the odds ratio 189
VI The Informed Guess - An Introduction to Confidence Intervals 191
14 Estimating the value of a single population parameter - the idea of confidence intervals 193
Confidence interval estimation for a population mean 194
The standard error of the mean 195
How we use the standard error of the mean to calculate a confidence interval for a population mean 197
Confidence interval for a population proportion 200
Estimating a confidence interval for the median of a single population 203
15 Using confidence intervals to compare two population parameters 206
What's the difference? 207
Comparing two independent population means 207
An example using birthweights 208
Assessing the evidence using the confidence interval 211
Comparing two paired population means 215
Within-subject and between-subject variations 215
Comparing two independent population proportions 217
Comparing two independent population medians - the Mann-Whitney rank sums method 219
Comparing two matched population medians - the Wilcoxon signed-ranks method 220
16 Confidence intervals for the ratio of two population parameters 224
Getting a confidence interval for the ratio of two independent population means 225
Confidence interval for a population risk ratio 226
Confidence intervals for a population odds ratio 229
Confidence intervals for hazard ratios 232
VII Putting it to the Test 235
17 Testing hypotheses about the difference between two population parameters 237
Answering the question 238
The hypothesis 238
The null hypothesis 239
The hypothesis testing process 240
The p-value and the decision rule 241
A brief summary of a few of the commonest tests 242
Using the p-value to compare the means of two independent populations 244
Interpreting computer hypothesis test results for the difference in two independent population means - the two-sample t test 245
Output from Minitab - two-sample t test of difference in mean birthweights of babies born to white mothers and to non-white mothers 245
Output from SPSS_: two-sample t test of difference in mean birthweights of babies born to white mothers and to non-white mothers 246
Comparing the means of two paired populations - the matched-pairs t test 248
Using p-values to compare the medians of two independent populations: the Mann-Whitney rank-sums test 248
How the Mann-Whitney test works 249
Correction for multiple comparisons 250
The Bonferroni correction for multiple testing 250
Interpreting computer output for the Mann-Whitney test 252
With Minitab 252
With SPSS 252
Two matched medians - the Wilcoxon signed-ranks test 254
Confidence intervals versus hypothesis testing 254
What could possibly go wrong? 255
Types of error 256
The power of a test 257
Maximising power - calculating sample size 258
Rule of thumb 1. Comparing the means of two independent populations (metric data) 258
Rule of thumb 2. Comparing the proportions of two independent populations (binary data) 259
18 The Chi-squared (?2) test - what, why, and how? 261
Of all the tests in all the world - you had to walk into my hypothesis testing procedure 262
Using chi-squared to test for related-ness or for the equality of proportions 262
Calculating the chi-squared statistic 265
Using the chi-squared statistic 267
Yate's correction (continuity correction) 268
Fisher's exact test 268
The chi-squared test with Minitab 269
The chi-squared test with SPSS 270
The chi-squared test for trend 272
SPSS output for chi-squared trend test 274
19 Testing hypotheses about the ratio of two population parameters 276
Preamble 276
The chi-squared test with the risk ratio 277
The chi-squared test with odds ratios 279
The chi-squared test with hazard ratios 281
VIII Becoming Acquainted 283
20 Measuring the association between two variables 285
Preamble - plotting data 286
Association 287
The scatterplot 287
The correlation coefficient 290
Pearson's correlation coefficient 290
Is the correlation coefficient statistically significant in the population? 292
Spearman's rank correlation coefficient 294
21 Measuring agreement 298
To agree or not agree: that is the question 298
Cohen's kappa (?) 300
Some shortcomings of kappa 303
Weighted kappa 303
Measuring the agreement between two metric continuous variables, the Bland-Altmann plot 303
IX Getting into a Relationship 307
22 Straight line models: linear regression 309
Health warning! 310
Relationship and association 310
A causal relationship - explaining variation 312
Refresher - finding the equation of a straight line from a graph 313
The linear regression model 314
First, is the relationship linear? 315
Estimating the regression parameters - the method of ordinary least squares (OLS) 316
Basic assumptions of the ordinary least squares procedure 317
Back to the example - is the relationship statistically significant? 318
Using SPSS to regress birthweight on mother's weight 318
Using Minitab 319
Interpreting the regression coefficients 320
Goodness-of-fit, R2 320
Multiple linear regression 322
Adjusted goodness-of-fit: R?2 324
Including nominal covariates in the regression model: design variables and coding 326
Building your model. Which variables to include? 327
Automated variable selection methods 328
Manual variable selection methods 329
Adjustment and confounding 330
Diagnostics - checking the basic assumptions of the multiple linear regression model 332
Analysis of variance 333
23 Curvy models: logistic regression 334
A second health warning! 335
The binary outcome variable 335
Finding an appropriate model when the outcome variable is binary 335
The logistic regression model 337
Estimating the parameter values 338
Interpreting the regression coefficients 338
Have we got a significant result? statistical inference in the logistic regression model 340
The Odds Ratio 341
The multiple logistic regression model 343
Building the model 344
Goodness-of-fit 346
24 Counting models: Poisson regression 349
Preamble 350
Poisson regression 350
The Poisson regression equation 351
Estimating ?1 and ?2 with the estimators b0 and b1 352
Interpreting the estimated coefficients of a Poisson regression, b0 and b1 352
Model building - variable selection 355
Goodness-of-fit 357
Zero-inflated Poisson regression 358
Negative binomial regression 359
Zero-inflated negative binomial regression 361
X Four More Chapters 363
25 Measuring survival 365
Preamble 366
Censored data 366
A simple example of survival in a single group 366
Calculating survival probabilities and the proportion surviving: the Kaplan-Meier table 368
The Kaplan-Meier curve 369
Determining median survival time 369
Comparing survival with two groups 370
The log-rank test 371
An example of the log-rank test in practice 372
The hazard ratio 372
The proportional hazards (Cox's) regression model - introduction 373
The proportional hazards (Cox's) regression model - the detail 376
Checking the assumptions of the proportional hazards model 377
An example of proportional hazards regression 377
26 Systematic review and meta-analysis 380
Introduction 381
Systematic review 381
The forest plot 383
Publication and other biases 384
The funnel plot 386
Significance tests for bias - Begg's and Egger's tests 387
Combining the studies: meta-analysis 389
The problem of heterogeneity - the Q and I2 tests 389
27 Diagnostic testing 393
Preamble 393
The measures - sensitivity and specificity 394
The positive prediction and negative prediction values (PPV and NPV) 395
The sensitivity-specificity trade-off 396
Using the ROC curve to find the optimal sensitivity versus specificity trade-off 397
28 Missing data 400
The missing data problem 400
Types of missing data 403
Missing completely at random (MCAR) 403
Missing at Random (MAR) 403
Missing not at random (MNAR) 404
Consequences of missing data 405
Dealing with missing data 405
Do nothing - the wing and prayer approach 406
List-wise deletion 406
Pair-wise deletion 407
Imputation methods - simple imputation 408
Replacement by the Mean 408
Last observation carried forward 409
Regression-based imputation 410
Multiple imputation 411
Full Information Maximum Likelihood (FIML) and other methods 412
Appendix: Table of random numbers 414
References 415
Solutions to Exercises 424
Index 457
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<p>medical statistics; funnel plot; scatterplot, medicine; statistics; medical data; medical charts; clinical trial; health care; epidemiology; data collection; sampling; confidence interval; hypothesis testing; regression; modelling; survival analysis; systematic review; meta-analysis; diagnostic testing</p>
Preface to the 4th Edition xix
Preface to the 3rd Edition xxi
Preface to the 2nd Edition xxiii
Preface to the 1st Edition xxv
Introduction xxvii
I Some Fundamental Stuff 1
1 First things first - the nature of data 3
Variables and data 3
Where are we going ...? 5
The good, the bad, and the ugly - types of variables 5
Categorical data 6
Nominal categorical data 6
Ordinal categorical data 7
Metric data 8
Discrete metric data 8
Continuous metric data 9
How can I tell what type of variable I am dealing with? 10
The baseline table 11
II Descriptive Statistics 15
2 Describing data with tables 17
Descriptive statistics. What can we do with raw data? 18
Frequency tables - nominal data 18
The frequency distribution 19
Relative frequency 20
Frequency tables - ordinal data 20
Frequency tables - metric data 22
Frequency tables with discrete metric data 22
Cumulative frequency 24
Frequency tables with continuous metric data - grouping the raw data 25
Open-ended groups 27
Cross-tabulation - contingency tables 28
Ranking data 30
3 Every picture tells a story - describing data with charts 31
Picture it! 32
Charting nominal and ordinal data 32
The pie chart 32
The simple bar chart 34
The clustered bar chart 35
The stacked bar chart 37
Charting discrete metric data 39
Charting continuous metric data 39
The histogram 39
The box (and whisker) plot 42
Charting cumulative data 44
The cumulative frequency curve with discrete metric data 44
The cumulative frequency curve with continuous metric data 44
Charting time-based data - the time series chart 47
The scatterplot 48
The bubbleplot 49
4 Describing data from its shape 51
The shape of things to come 51
Skewness and kurtosis as measures of shape 52
Kurtosis 55
Symmetric or mound-shaped distributions 56
Normalness - the Normal distribution 56
Bimodal distributions 58
Determining skew from a box plot 59
5 Measures of location - Numbers R us 62
Numbers, percentages, and proportions 62
Preamble 63
N umbers, percentages, and proportions 64
Handling percentages - for those of us who might need a reminder 65
Summary measures of location 67
The mode 68
The median 69
The mean 70
Percentiles 71
Calculating a percentile value 72
What is the most appropriate measure of location? 73
6 Measures of spread - Numbers R us - (again) 75
Preamble 76
The range 76
The interquartile range (IQR) 76
Estimating the median and interquartile range from the cumulative frequency curve 77
The boxplot (also known as the box and whisker plot) 79
Standard deviation 82
Standard deviation and the Normal distribution 84
Testing for Normality 86
Using SPSS 86
Using Minitab 87
Transforming data 88
7 Incidence, prevalence, and standardisation 92
Preamble 93
The incidence rate and the incidence rate ratio (IRR) 93
The incidence rate ratio 94
Prevalence 94
A couple of difficulties with measuring incidence and prevalence 97
Some other useful rates 97
Crude mortality rate 97
Case fatality rate 98
Crude maternal mortality rate 99
Crude birth rate 99
Attack rate 99
Age-specific mortality rate 99
Standardisation - the age-standardised mortality rate 101
The direct method 102
The standard population and the comparative mortality ratio (CMR) 103
The indirect method 106
The standardised mortality rate 107
III The Confounding Problem 111
8 Confounding - like the poor, (nearly) always with us 113
Preamble 114
What is confounding? 114
Confounding by indication 117
Residual confounding 119
Detecting confounding 119
Dealing with confounding - if confounding is such a problem, what can we do about it? 120
Using restriction 120
Using matching 121
Frequency matching 121
One-to-one matching 121
Using stratification 122
Using adjustment 122
Using randomisation 122
IV Design and Data 125
9 Research design - Part I: Observational study designs 127
Preamble 128
Hey ho! Hey ho! it's off to work we go 129
Types of study 129
Observational studies 130
Case reports 130
Case series studies 131
Cross-sectional studies 131
Descriptive cross-sectional studies 132
Confounding in descriptive cross-sectional studies 132
Analytic cross-sectional studies 133
Confounding in analytic cross-sectional studies 134
From here to eternity - cohort studies 135
Confounding in the cohort study design 139
Back to the future - case-control studies 139
Confounding in the case-control study design 141
Another example of a case-control study 142
Comparing cohort and case-control designs 143
Ecological studies 144
The ecological fallacy 145
10 Research design - Part II: getting stuck in - experimental studies 146
Clinical trials 147
Randomisation and the randomised controlled trial (RCT) 148
Block randomisation 149
Stratification 149
Blinding 149
The crossover RCT 150
Selection of participants for an RCT 153
Intention to treat analysis (ITT) 154
11 Getting the participants for your study: ways of sampling 156
From populations to samples - statistical inference 157
Collecting the data - types of sample 158
The simple random sample and its offspring 159
The systematic random sample 159
The stratified random sample 160
The cluster sample 160
Consecutive and convenience samples 161
How many participants should we have? Sample size 162
Inclusion and exclusion criteria 162
Getting the data 163
V Chance Would Be a Fine Thing 165
12 The idea of probability 167
Preamble 167
Calculating probability - proportional frequency 168
Two useful rules for simple probability 169
Rule 1. The multiplication rule for independent events 169
Rule 2. The addition rule for mutually exclusive events 170
Conditional and Bayesian statistics 171
Probability distributions 171
Discrete versus continuous probability distributions 172
The binomial probability distribution 172
The Poisson probability distribution 173
The Normal probability distribution 174
13 Risk and odds 175
Absolute risk and the absolute risk reduction (ARR) 176
The risk ratio 178
The reduction in the risk ratio (or relative risk reduction (RRR)) 178
A general formula for the risk ratio 179
Reference value 179
N umber needed to treat (NNT) 180
What happens if the initial risk is small? 181
Confounding with the risk ratio 182
Odds 183
Why you can't calculate risk in a case-control study 185
The link between probability and odds 186
The odds ratio 186
Confounding with the odds ratio 189
Approximating the risk ratio from the odds ratio 189
VI The Informed Guess - An Introduction to Confidence Intervals 191
14 Estimating the value of a single population parameter - the idea of confidence intervals 193
Confidence interval estimation for a population mean 194
The standard error of the mean 195
How we use the standard error of the mean to calculate a confidence interval for a population mean 197
Confidence interval for a population proportion 200
Estimating a confidence interval for the median of a single population 203
15 Using confidence intervals to compare two population parameters 206
What's the difference? 207
Comparing two independent population means 207
An example using birthweights 208
Assessing the evidence using the confidence interval 211
Comparing two paired population means 215
Within-subject and between-subject variations 215
Comparing two independent population proportions 217
Comparing two independent population medians - the Mann-Whitney rank sums method 219
Comparing two matched population medians - the Wilcoxon signed-ranks method 220
16 Confidence intervals for the ratio of two population parameters 224
Getting a confidence interval for the ratio of two independent population means 225
Confidence interval for a population risk ratio 226
Confidence intervals for a population odds ratio 229
Confidence intervals for hazard ratios 232
VII Putting it to the Test 235
17 Testing hypotheses about the difference between two population parameters 237
Answering the question 238
The hypothesis 238
The null hypothesis 239
The hypothesis testing process 240
The p-value and the decision rule 241
A brief summary of a few of the commonest tests 242
Using the p-value to compare the means of two independent populations 244
Interpreting computer hypothesis test results for the difference in two independent population means - the two-sample t test 245
Output from Minitab - two-sample t test of difference in mean birthweights of babies born to white mothers and to non-white mothers 245
Output from SPSS_: two-sample t test of difference in mean birthweights of babies born to white mothers and to non-white mothers 246
Comparing the means of two paired populations - the matched-pairs t test 248
Using p-values to compare the medians of two independent populations: the Mann-Whitney rank-sums test 248
How the Mann-Whitney test works 249
Correction for multiple comparisons 250
The Bonferroni correction for multiple testing 250
Interpreting computer output for the Mann-Whitney test 252
With Minitab 252
With SPSS 252
Two matched medians - the Wilcoxon signed-ranks test 254
Confidence intervals versus hypothesis testing 254
What could possibly go wrong? 255
Types of error 256
The power of a test 257
Maximising power - calculating sample size 258
Rule of thumb 1. Comparing the means of two independent populations (metric data) 258
Rule of thumb 2. Comparing the proportions of two independent populations (binary data) 259
18 The Chi-squared (?2) test - what, why, and how? 261
Of all the tests in all the world - you had to walk into my hypothesis testing procedure 262
Using chi-squared to test for related-ness or for the equality of proportions 262
Calculating the chi-squared statistic 265
Using the chi-squared statistic 267
Yate's correction (continuity correction) 268
Fisher's exact test 268
The chi-squared test with Minitab 269
The chi-squared test with SPSS 270
The chi-squared test for trend 272
SPSS output for chi-squared trend test 274
19 Testing hypotheses about the ratio of two population parameters 276
Preamble 276
The chi-squared test with the risk ratio 277
The chi-squared test with odds ratios 279
The chi-squared test with hazard ratios 281
VIII Becoming Acquainted 283
20 Measuring the association between two variables 285
Preamble - plotting data 286
Association 287
The scatterplot 287
The correlation coefficient 290
Pearson's correlation coefficient 290
Is the correlation coefficient statistically significant in the population? 292
Spearman's rank correlation coefficient 294
21 Measuring agreement 298
To agree or not agree: that is the question 298
Cohen's kappa (?) 300
Some shortcomings of kappa 303
Weighted kappa 303
Measuring the agreement between two metric continuous variables, the Bland-Altmann plot 303
IX Getting into a Relationship 307
22 Straight line models: linear regression 309
Health warning! 310
Relationship and association 310
A causal relationship - explaining variation 312
Refresher - finding the equation of a straight line from a graph 313
The linear regression model 314
First, is the relationship linear? 315
Estimating the regression parameters - the method of ordinary least squares (OLS) 316
Basic assumptions of the ordinary least squares procedure 317
Back to the example - is the relationship statistically significant? 318
Using SPSS to regress birthweight on mother's weight 318
Using Minitab 319
Interpreting the regression coefficients 320
Goodness-of-fit, R2 320
Multiple linear regression 322
Adjusted goodness-of-fit: R?2 324
Including nominal covariates in the regression model: design variables and coding 326
Building your model. Which variables to include? 327
Automated variable selection methods 328
Manual variable selection methods 329
Adjustment and confounding 330
Diagnostics - checking the basic assumptions of the multiple linear regression model 332
Analysis of variance 333
23 Curvy models: logistic regression 334
A second health warning! 335
The binary outcome variable 335
Finding an appropriate model when the outcome variable is binary 335
The logistic regression model 337
Estimating the parameter values 338
Interpreting the regression coefficients 338
Have we got a significant result? statistical inference in the logistic regression model 340
The Odds Ratio 341
The multiple logistic regression model 343
Building the model 344
Goodness-of-fit 346
24 Counting models: Poisson regression 349
Preamble 350
Poisson regression 350
The Poisson regression equation 351
Estimating ?1 and ?2 with the estimators b0 and b1 352
Interpreting the estimated coefficients of a Poisson regression, b0 and b1 352
Model building - variable selection 355
Goodness-of-fit 357
Zero-inflated Poisson regression 358
Negative binomial regression 359
Zero-inflated negative binomial regression 361
X Four More Chapters 363
25 Measuring survival 365
Preamble 366
Censored data 366
A simple example of survival in a single group 366
Calculating survival probabilities and the proportion surviving: the Kaplan-Meier table 368
The Kaplan-Meier curve 369
Determining median survival time 369
Comparing survival with two groups 370
The log-rank test 371
An example of the log-rank test in practice 372
The hazard ratio 372
The proportional hazards (Cox's) regression model - introduction 373
The proportional hazards (Cox's) regression model - the detail 376
Checking the assumptions of the proportional hazards model 377
An example of proportional hazards regression 377
26 Systematic review and meta-analysis 380
Introduction 381
Systematic review 381
The forest plot 383
Publication and other biases 384
The funnel plot 386
Significance tests for bias - Begg's and Egger's tests 387
Combining the studies: meta-analysis 389
The problem of heterogeneity - the Q and I2 tests 389
27 Diagnostic testing 393
Preamble 393
The measures - sensitivity and specificity 394
The positive prediction and negative prediction values (PPV and NPV) 395
The sensitivity-specificity trade-off 396
Using the ROC curve to find the optimal sensitivity versus specificity trade-off 397
28 Missing data 400
The missing data problem 400
Types of missing data 403
Missing completely at random (MCAR) 403
Missing at Random (MAR) 403
Missing not at random (MNAR) 404
Consequences of missing data 405
Dealing with missing data 405
Do nothing - the wing and prayer approach 406
List-wise deletion 406
Pair-wise deletion 407
Imputation methods - simple imputation 408
Replacement by the Mean 408
Last observation carried forward 409
Regression-based imputation 410
Multiple imputation 411
Full Information Maximum Likelihood (FIML) and other methods 412
Appendix: Table of random numbers 414
References 415
Solutions to Exercises 424
Index 457
Preface to the 3rd Edition xxi
Preface to the 2nd Edition xxiii
Preface to the 1st Edition xxv
Introduction xxvii
I Some Fundamental Stuff 1
1 First things first - the nature of data 3
Variables and data 3
Where are we going ...? 5
The good, the bad, and the ugly - types of variables 5
Categorical data 6
Nominal categorical data 6
Ordinal categorical data 7
Metric data 8
Discrete metric data 8
Continuous metric data 9
How can I tell what type of variable I am dealing with? 10
The baseline table 11
II Descriptive Statistics 15
2 Describing data with tables 17
Descriptive statistics. What can we do with raw data? 18
Frequency tables - nominal data 18
The frequency distribution 19
Relative frequency 20
Frequency tables - ordinal data 20
Frequency tables - metric data 22
Frequency tables with discrete metric data 22
Cumulative frequency 24
Frequency tables with continuous metric data - grouping the raw data 25
Open-ended groups 27
Cross-tabulation - contingency tables 28
Ranking data 30
3 Every picture tells a story - describing data with charts 31
Picture it! 32
Charting nominal and ordinal data 32
The pie chart 32
The simple bar chart 34
The clustered bar chart 35
The stacked bar chart 37
Charting discrete metric data 39
Charting continuous metric data 39
The histogram 39
The box (and whisker) plot 42
Charting cumulative data 44
The cumulative frequency curve with discrete metric data 44
The cumulative frequency curve with continuous metric data 44
Charting time-based data - the time series chart 47
The scatterplot 48
The bubbleplot 49
4 Describing data from its shape 51
The shape of things to come 51
Skewness and kurtosis as measures of shape 52
Kurtosis 55
Symmetric or mound-shaped distributions 56
Normalness - the Normal distribution 56
Bimodal distributions 58
Determining skew from a box plot 59
5 Measures of location - Numbers R us 62
Numbers, percentages, and proportions 62
Preamble 63
N umbers, percentages, and proportions 64
Handling percentages - for those of us who might need a reminder 65
Summary measures of location 67
The mode 68
The median 69
The mean 70
Percentiles 71
Calculating a percentile value 72
What is the most appropriate measure of location? 73
6 Measures of spread - Numbers R us - (again) 75
Preamble 76
The range 76
The interquartile range (IQR) 76
Estimating the median and interquartile range from the cumulative frequency curve 77
The boxplot (also known as the box and whisker plot) 79
Standard deviation 82
Standard deviation and the Normal distribution 84
Testing for Normality 86
Using SPSS 86
Using Minitab 87
Transforming data 88
7 Incidence, prevalence, and standardisation 92
Preamble 93
The incidence rate and the incidence rate ratio (IRR) 93
The incidence rate ratio 94
Prevalence 94
A couple of difficulties with measuring incidence and prevalence 97
Some other useful rates 97
Crude mortality rate 97
Case fatality rate 98
Crude maternal mortality rate 99
Crude birth rate 99
Attack rate 99
Age-specific mortality rate 99
Standardisation - the age-standardised mortality rate 101
The direct method 102
The standard population and the comparative mortality ratio (CMR) 103
The indirect method 106
The standardised mortality rate 107
III The Confounding Problem 111
8 Confounding - like the poor, (nearly) always with us 113
Preamble 114
What is confounding? 114
Confounding by indication 117
Residual confounding 119
Detecting confounding 119
Dealing with confounding - if confounding is such a problem, what can we do about it? 120
Using restriction 120
Using matching 121
Frequency matching 121
One-to-one matching 121
Using stratification 122
Using adjustment 122
Using randomisation 122
IV Design and Data 125
9 Research design - Part I: Observational study designs 127
Preamble 128
Hey ho! Hey ho! it's off to work we go 129
Types of study 129
Observational studies 130
Case reports 130
Case series studies 131
Cross-sectional studies 131
Descriptive cross-sectional studies 132
Confounding in descriptive cross-sectional studies 132
Analytic cross-sectional studies 133
Confounding in analytic cross-sectional studies 134
From here to eternity - cohort studies 135
Confounding in the cohort study design 139
Back to the future - case-control studies 139
Confounding in the case-control study design 141
Another example of a case-control study 142
Comparing cohort and case-control designs 143
Ecological studies 144
The ecological fallacy 145
10 Research design - Part II: getting stuck in - experimental studies 146
Clinical trials 147
Randomisation and the randomised controlled trial (RCT) 148
Block randomisation 149
Stratification 149
Blinding 149
The crossover RCT 150
Selection of participants for an RCT 153
Intention to treat analysis (ITT) 154
11 Getting the participants for your study: ways of sampling 156
From populations to samples - statistical inference 157
Collecting the data - types of sample 158
The simple random sample and its offspring 159
The systematic random sample 159
The stratified random sample 160
The cluster sample 160
Consecutive and convenience samples 161
How many participants should we have? Sample size 162
Inclusion and exclusion criteria 162
Getting the data 163
V Chance Would Be a Fine Thing 165
12 The idea of probability 167
Preamble 167
Calculating probability - proportional frequency 168
Two useful rules for simple probability 169
Rule 1. The multiplication rule for independent events 169
Rule 2. The addition rule for mutually exclusive events 170
Conditional and Bayesian statistics 171
Probability distributions 171
Discrete versus continuous probability distributions 172
The binomial probability distribution 172
The Poisson probability distribution 173
The Normal probability distribution 174
13 Risk and odds 175
Absolute risk and the absolute risk reduction (ARR) 176
The risk ratio 178
The reduction in the risk ratio (or relative risk reduction (RRR)) 178
A general formula for the risk ratio 179
Reference value 179
N umber needed to treat (NNT) 180
What happens if the initial risk is small? 181
Confounding with the risk ratio 182
Odds 183
Why you can't calculate risk in a case-control study 185
The link between probability and odds 186
The odds ratio 186
Confounding with the odds ratio 189
Approximating the risk ratio from the odds ratio 189
VI The Informed Guess - An Introduction to Confidence Intervals 191
14 Estimating the value of a single population parameter - the idea of confidence intervals 193
Confidence interval estimation for a population mean 194
The standard error of the mean 195
How we use the standard error of the mean to calculate a confidence interval for a population mean 197
Confidence interval for a population proportion 200
Estimating a confidence interval for the median of a single population 203
15 Using confidence intervals to compare two population parameters 206
What's the difference? 207
Comparing two independent population means 207
An example using birthweights 208
Assessing the evidence using the confidence interval 211
Comparing two paired population means 215
Within-subject and between-subject variations 215
Comparing two independent population proportions 217
Comparing two independent population medians - the Mann-Whitney rank sums method 219
Comparing two matched population medians - the Wilcoxon signed-ranks method 220
16 Confidence intervals for the ratio of two population parameters 224
Getting a confidence interval for the ratio of two independent population means 225
Confidence interval for a population risk ratio 226
Confidence intervals for a population odds ratio 229
Confidence intervals for hazard ratios 232
VII Putting it to the Test 235
17 Testing hypotheses about the difference between two population parameters 237
Answering the question 238
The hypothesis 238
The null hypothesis 239
The hypothesis testing process 240
The p-value and the decision rule 241
A brief summary of a few of the commonest tests 242
Using the p-value to compare the means of two independent populations 244
Interpreting computer hypothesis test results for the difference in two independent population means - the two-sample t test 245
Output from Minitab - two-sample t test of difference in mean birthweights of babies born to white mothers and to non-white mothers 245
Output from SPSS_: two-sample t test of difference in mean birthweights of babies born to white mothers and to non-white mothers 246
Comparing the means of two paired populations - the matched-pairs t test 248
Using p-values to compare the medians of two independent populations: the Mann-Whitney rank-sums test 248
How the Mann-Whitney test works 249
Correction for multiple comparisons 250
The Bonferroni correction for multiple testing 250
Interpreting computer output for the Mann-Whitney test 252
With Minitab 252
With SPSS 252
Two matched medians - the Wilcoxon signed-ranks test 254
Confidence intervals versus hypothesis testing 254
What could possibly go wrong? 255
Types of error 256
The power of a test 257
Maximising power - calculating sample size 258
Rule of thumb 1. Comparing the means of two independent populations (metric data) 258
Rule of thumb 2. Comparing the proportions of two independent populations (binary data) 259
18 The Chi-squared (?2) test - what, why, and how? 261
Of all the tests in all the world - you had to walk into my hypothesis testing procedure 262
Using chi-squared to test for related-ness or for the equality of proportions 262
Calculating the chi-squared statistic 265
Using the chi-squared statistic 267
Yate's correction (continuity correction) 268
Fisher's exact test 268
The chi-squared test with Minitab 269
The chi-squared test with SPSS 270
The chi-squared test for trend 272
SPSS output for chi-squared trend test 274
19 Testing hypotheses about the ratio of two population parameters 276
Preamble 276
The chi-squared test with the risk ratio 277
The chi-squared test with odds ratios 279
The chi-squared test with hazard ratios 281
VIII Becoming Acquainted 283
20 Measuring the association between two variables 285
Preamble - plotting data 286
Association 287
The scatterplot 287
The correlation coefficient 290
Pearson's correlation coefficient 290
Is the correlation coefficient statistically significant in the population? 292
Spearman's rank correlation coefficient 294
21 Measuring agreement 298
To agree or not agree: that is the question 298
Cohen's kappa (?) 300
Some shortcomings of kappa 303
Weighted kappa 303
Measuring the agreement between two metric continuous variables, the Bland-Altmann plot 303
IX Getting into a Relationship 307
22 Straight line models: linear regression 309
Health warning! 310
Relationship and association 310
A causal relationship - explaining variation 312
Refresher - finding the equation of a straight line from a graph 313
The linear regression model 314
First, is the relationship linear? 315
Estimating the regression parameters - the method of ordinary least squares (OLS) 316
Basic assumptions of the ordinary least squares procedure 317
Back to the example - is the relationship statistically significant? 318
Using SPSS to regress birthweight on mother's weight 318
Using Minitab 319
Interpreting the regression coefficients 320
Goodness-of-fit, R2 320
Multiple linear regression 322
Adjusted goodness-of-fit: R?2 324
Including nominal covariates in the regression model: design variables and coding 326
Building your model. Which variables to include? 327
Automated variable selection methods 328
Manual variable selection methods 329
Adjustment and confounding 330
Diagnostics - checking the basic assumptions of the multiple linear regression model 332
Analysis of variance 333
23 Curvy models: logistic regression 334
A second health warning! 335
The binary outcome variable 335
Finding an appropriate model when the outcome variable is binary 335
The logistic regression model 337
Estimating the parameter values 338
Interpreting the regression coefficients 338
Have we got a significant result? statistical inference in the logistic regression model 340
The Odds Ratio 341
The multiple logistic regression model 343
Building the model 344
Goodness-of-fit 346
24 Counting models: Poisson regression 349
Preamble 350
Poisson regression 350
The Poisson regression equation 351
Estimating ?1 and ?2 with the estimators b0 and b1 352
Interpreting the estimated coefficients of a Poisson regression, b0 and b1 352
Model building - variable selection 355
Goodness-of-fit 357
Zero-inflated Poisson regression 358
Negative binomial regression 359
Zero-inflated negative binomial regression 361
X Four More Chapters 363
25 Measuring survival 365
Preamble 366
Censored data 366
A simple example of survival in a single group 366
Calculating survival probabilities and the proportion surviving: the Kaplan-Meier table 368
The Kaplan-Meier curve 369
Determining median survival time 369
Comparing survival with two groups 370
The log-rank test 371
An example of the log-rank test in practice 372
The hazard ratio 372
The proportional hazards (Cox's) regression model - introduction 373
The proportional hazards (Cox's) regression model - the detail 376
Checking the assumptions of the proportional hazards model 377
An example of proportional hazards regression 377
26 Systematic review and meta-analysis 380
Introduction 381
Systematic review 381
The forest plot 383
Publication and other biases 384
The funnel plot 386
Significance tests for bias - Begg's and Egger's tests 387
Combining the studies: meta-analysis 389
The problem of heterogeneity - the Q and I2 tests 389
27 Diagnostic testing 393
Preamble 393
The measures - sensitivity and specificity 394
The positive prediction and negative prediction values (PPV and NPV) 395
The sensitivity-specificity trade-off 396
Using the ROC curve to find the optimal sensitivity versus specificity trade-off 397
28 Missing data 400
The missing data problem 400
Types of missing data 403
Missing completely at random (MCAR) 403
Missing at Random (MAR) 403
Missing not at random (MNAR) 404
Consequences of missing data 405
Dealing with missing data 405
Do nothing - the wing and prayer approach 406
List-wise deletion 406
Pair-wise deletion 407
Imputation methods - simple imputation 408
Replacement by the Mean 408
Last observation carried forward 409
Regression-based imputation 410
Multiple imputation 411
Full Information Maximum Likelihood (FIML) and other methods 412
Appendix: Table of random numbers 414
References 415
Solutions to Exercises 424
Index 457
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<p>medical statistics; funnel plot; scatterplot, medicine; statistics; medical data; medical charts; clinical trial; health care; epidemiology; data collection; sampling; confidence interval; hypothesis testing; regression; modelling; survival analysis; systematic review; meta-analysis; diagnostic testing</p>
