Medical Biostatistics, Fourth Edition

Table S.1

Methods to Compute Some Confidence Intervals

Parameter of Interest

Conditions

95% CI

Page Number

Proportion (π)

Large n, 0 <p< 1

Equation 12.11

264

Small n, any p

Figure 12.5

272

Any n, p = 0 or 1 (confidence bound)

Table 12.4

273

Mean (μ)

Large n, σ known, almost any underlying distribution

Equation 12.14

266

Small n, σ known, underlying Gaussian

Equation 12.14

266

Small n, σ known or unknown, underlying nonGaussian

Table 12.5 (CI for median)

275

Any n, σ unknown, underlying Gaussian

Equation 12.15

266

Large n, σ unknown, underlying nonGaussian

Equation 12.15

266

Median

Gaussian distribution

Equation 12.18b

268

NonGaussian conditions

Table 12.5

275

Difference in proportions (π1 – π2)

Large n1, n2 – Independent samples

Equation 12.20

269

Large n1, n2 – Paired samples

Equation 12.23

271

Difference in means (μ1 – μ2) (σ unknown)

Independent samples

Large n1, n2 – Almost any underlying distribution

Equation 12.21

269

Small n1, n2 – Underlying Gaussian

Equation 12.21

269

Paired samples

Same as for one sample after taking the difference

Relative risk

Large n1, n2 – Independent samples

Equation 14.4

351

Large n1, n2 – Paired samples

Same as for OR when prevalence is small (Equation 14.21)

351

Attributable risk

Large n1, n2 – Independent samples

Same as for (π1 – π2)

Large n1, n2 – Paired samples

Equation 14.12

355

Number needed to treat

Large n1, n2 – Independent samples

Section 14.1.3.3

355

Odds ratio

Large n1, n2 – Independent samples

Equation 14.18

360

Large n1, n2 – Paired samples

Equation 14.21

362

Regression coefficient and intercept (simple linear)

Large n (Gaussian residuals)

Use Equation 16.7

435

Regression line (confidence band)

Large n (Gaussian residuals)

Section 16.2.1.5

437

Logistic coefficient

Large n

Section 17.2.2

480

Median effective dose

Up-and-down trial

Section 15.1.4

386

Table S.2

Statistical Procedures for Test of Hypothesis on Proportions

Parameter of Interest and Setup

Conditions

Main Criterion

Equation/

Section

Page Number

One-way and 2×2 Tables

One dichotomous variable

Independent trials

Any n

Binomial

Use Equation 13.1

308

Large n

Gaussian Z

Equation 13.3

309

One polytomous variable

Independent trials

Large n

Goodness-of-fit chi-square

Equation 13.5

312

Small n

Multinomial

Use Equation 13.6

316

Two dichotomous variables (2 × 2)

Two independent samples

Large n

Chi-square or Gaussian Z

Equation 13.8 or 13.9

319/320

Small n

Fisher exact

Equation 13.11

326

Detecting a medically important difference – Large n

Gaussian Z

Equation 13.10

321

Equivalence, superiority and noninferiority tests

TOSTs and others

Table 13.10

325

Matched pairs

Large n

McNemar

Equation 13.12

328

Small n

Binomial

Equation 13.13

329

Crossover design

Large n

Chi-square

Section 13.2.2.5

322

Small n

Fisher exact

Equation 13.11

326

Bigger Tables,

No Matching

Large n Required

The Case of Smalln Not Discussed in This Book

Association (nominal)

2×C tables

Chi-square

Equation 13.15

332

Trend in proportions (ordinal)

2×C tables

Chi-square for trend

Equation 13.16

333

Dichotomy in repeated measures

Many related 2×2 tables

Cochran Q

Equation 13.18

324

Association

R×C tables

Chi-square

Equation 13.15

332

Association

Three-way tables

Test of full independence

Chi-square

Equation 13.19

340

Test of other types of independence (log–linear models)

G2

Three-way extension of Equation 13.22

341

I × Itable

Matched pairs

McNemar–Bowker

Section 13.3.2.2

337

Stratified

Stratified into many 2×2 tables

Mantel–Haenszel chi-square

Equation 14.26

365

Table S.3

Procedures for Test of Hypothesis on Relative Risk (RR) and Odds Ratio (OR)

Parameter of Interest and Setup

Conditions

Main Criterion

Equation/

Section

Page Number

Relative and Attributable Risks

Large n Required

The Case of Small n Not Discussed in This Book

ln(RR)

Two independent samples

Gaussian Z or chi-square

Equation 14.5 or 13.8

353/320

RR

Matched pairs

As for OR (Gaussian Z or McNemar)

Equation 14.22 or 14.23

362

AR

Two independent samples

Chi-square or Gaussian Z

Equation 13.8 or 13.9

319/320

Matched pairs

McNemar

Equation 13.12

328

Odds Ratio

Large n Required

The Case of Small n Not Discussed in This Book

ln(OR)

Two independent samples

Chi-square

Equation 13.8

319

OR

Matched pairs

Gaussian Z or McNemar

Equation 14.22 or 14.23

363/363

Stratified

Mantel–Haenszel chi-square

Equation 14.26

365

Table S.4

Statistical Procedures for Test of Hypothesis on Means or Locations

Setu

Conditions

Main Criterion

Equation/Section

Page Number

One sample

Comparison with prespecified – Gaussian

σ known

Gaussian Z

Section 15.1.1.1

378

σ not known

Student t

Equation 15.1

378

Comparison of two groups

Paired – Gaussian

Student t

Equation 15.3

380

Paired – NonGaussian

Any n

Sign test

Equation 15.17a–c

402

5 ≤n≤ 19

Wilcoxon signed-ranks WS

Equation 15.18a

402

20 ≤n≤ 29

Standardized WS referred to Gaussian Z

Equation 15.18b

403

n≥ 30

Student t

Equation 15.1

378

Unpaired – Gaussian

Equal variances

Student t

Equation 15.6a

381

Unequal variances

Welch

Equation 15.6b

381

Unpaired – NonGaussian

n1, n2 between (4, 9)

Wilcoxon rank-sum WR

Equation 15.19

404

n1, n2 between (10, 29)

Standardized WR referred to Gaussian Z

Equation 15.20

405

n1, n2≥ 30

Student t

Equation 15.6a or 15.6b

381

Crossover design – Gaussian conditions

Student t

Section 15.1.3

383

Detecting medically important difference – Gaussian conditions

Student t

Equation 15.23

414

Equivalence tests – Gaussian conditions

Student t

Section 15.4.2.2

415

Comparison of three or more groups

One-way layout – Gaussian

ANOVA F

Equation 15.8

389

One-way layout – NonGaussian

n≤ 5

Kruskal–Wallis H

Equation 15.21

406

n≥ 6

H referred to chi-square

Equation 15.21

406

Two-way layout – Gaussian

ANOVA F

Section 15.2.2

392

Two-way layout – NonGaussian (one observation per cell – repeated measures)

J≤ 13 and K = 3

Friedman S

Equation 15.22a or 15.22b

408

J≤ 8 and K = 4

Friedman S

Equation 15.22a or 15.22b

408

J≤ 5 and K = 5

Friedman S

Equation 15.22a or 15.22b

408

Larger J, K

S referred to chi-square

Equation 15.22a or 15.22b

408

Multiple comparisons

Gaussian conditions

All pairwise

Tukey D

Equation 15.15

399

With control group

Dunnett

Equation 15.16

400

Few comparisons

Bonferroni

Section 15.2.4.1

399

Repeated measures

Gaussian conditions

F-test with Hyunh–Feldt correction

Section 15.2.3

399

Table S.5

Procedures for Test of Hypothesis on Some Other Parameters

Parameter of Interest and Setup

Conditions

Main Criterion

Equation/

Section

Page Number

One sample

Product–moment correlation

Gaussian conditions

Student t

Equation 16.20

455

Serial correlation

Gaussian conditions

Durbin–Watson

Section 16.3.4

442

Intraclass correlation

Gaussian conditions

F

Section 16.5.3.2

459

Spherecity (repeated measures)

Gaussian conditions

Mauchly

Section 15.2.3.2

459

Goodness of fit of whole model

Large n

Hosmer–Lemeshow

Section 17.1.2.3

474

Two-sample Comparison

Two independent samples

Comparison of two distributions

Very large n(mean and SD known)

Kolmogorov-Smirnov

Section 12.4.2.1

285

Large n

Shapiro–Wilk

Section 12.4.2.1

285

Moderate n

Anderson–Darling

Section 12.4.2.1

285

Comparison of two correlations

Gaussian  conditions

Fisher z-transformation

Section 16.5.1.3

456

Comparison of two survival curves

Any distribution

Log-rank

Section 18.3.1.2

519

Comparison of two variances

Gaussian conditions

F or Levene

Section 15.1.2.2

381

Table S.6

Methods for Studying the Nature of Relationship

Dependent Variable (y)

Independent Variables (xs)

Method

Equation/

Section

Page Number

Quantitativea

Qualitative

ANOVA

Section 15.2

387

Quantitative

Quantitative

Quantitative regression

Chapter 16

423

Quantitative^a

Mixture of qualitative and quantitative

ANCOVA

Section 16.4.2.4

452

Qualitative (dichotomous)

Qualitative or quantitative or mixture

Logistic

Sections 17.1

469

Qualitative (polytomous)

Qualitative or quantitative or mixture

Logistic – any two categories at a time

Section 17.3.2

483

Quantitative

Discriminant

Section 19.2.3

539

Survival probability

Groups – duration in time intervals

Life table

Equation 18.8

511

Groups – duration in continuous time

Kaplan–Meier

Equation 18.10

513

Hazard ratio

Mixture of qualitative and quantitative

Cox model

Section 18.3.2

521

Note:  Large n required, particularly for tests of significance. Exact methods for small n not discussed in this book.

a  Quantitative are variables on metric scale without any broad categories. Fine categories are admissible.

Table S.7

Main Methods of Measurement of Strength of Relationship between Two Variables

Types of Variables

Measure

Equation/Section

Page Number

Both qualitative

Binary categories

OR, Jaccard, Yule, and several others

Section 17.5.1.1

488

Polytomous categories – nominal

Phi-coefficient

Equation 17.7a

490

Tschuprow coefficient

Equation 17.7b

490

Contingency coefficient

Equation 17.7c

490

Cramer V

Equation 17.7d

490

Proportional reduction in error

Equation 17.8

492

Polytomous categories – ordinal

Kendall tau, Goodman–Kruskal gamma, Somerd

Section 17.5.1.4

492

Dependent qualitative and independent quantitative

Odds ratio

Section 17.1

469

Dependent quantitative and independent qualitative

R2 from ANOVA or η2 from regression

Equation 17.9 or Equation 16.11

17.9/440

Both quantitative

For multiple linear (more than 2 variables)

R2 from regression

Use Equation 16.12

441

For simple linear

Product–moment correlation (r)

Equation 16.19

452

For monotonic relation

Spearman (rank) correlation (rS)

Equation 16.22

457

For intraclass

Intraclass correlation (rI)

Equation 16.23

459

With previous value

Serial (auto) correlation

Section 16.5.1.4

456

Keeping other variables fixed

Partial correlation

Equation 16.21

456

Two sets of quantitative variables

Canonical correlation

Section 19.2.1.1

531

One quantitative and the other dichotomous

Biserial

Section 17.5.2.2

495

Agreement

Qualitative

Cohen kappa

Equation 17.10

496

Quantitative

Limits of disagreement

Section 16.6.1.2

461

Alternative methods

Section 16.6.2.1

463

Intraclass correlation

Equation 16.23

459

Table S.8

Multivariate Methods in Different Situations (Large n Required)

Nature of the Variables

Objective

Types of Variables

Statistical Method

Section

Page Number

A dependent set and an independent set

Relationship

Both quantitative

Multivariate multiple regression

Section 19.2.1.2

532

Equality of means of dependents

Dependent quantitative and independent qualitative

MANOVA

Section 19.2.2.1

537

Dependent is one (many groups)

Classify subjects into known groups

Independent quantitative

Discriminant analysis

Section 19.2.3.1

539

All variables interrelated (none is dependent)

Discover natural clusters of subjects

Qualitative or quantitative or mixed

Cluster analysis

Section 19.3.1

543