Medical Biostatistics

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From: Medical Biostatistics, Second Edition by A. Indrayan (Chapman & Hall/CRC Press, New York), 2008

Summary Tables for Elementary Statistical Methods

Last column refers to the equation/para/section of the book

Summary-1 Methods to compute some confidence intervals

Parameter of interest	Conditions		95% CI
Proportion (p)	(i) Large n, p ¹ 0 and p ¹ 1 (ii) Small n, any p (iii) Any n, p = 0 or 1 (bound)		Eq. (12.11) Figure 12-4 Table 12-4

Mean (m)	(i) Large n, s known, almost any underlying distribution		Eq. (12.14)
	(ii) Small n, s known or unknown, underlying non-Gaussian		Table 12-5 (CI for median)
	(iii) Any n, s unknown, underlying Gaussian		Eq. (12.15)
	(iv) Large n, s unknown, underlying non-Gaussian		Eq. (12.15)
	(v) Small n, s known, underlying Gaussian		Eq. (12.14)

Difference (p₁– p₂)	(i) Large n₁, n₂—Independent sample		Eq. (12.20)
	(ii) Large n₁, n₂—Paired samples		Eq. (12.23)
	(iii) Small n₁, n₂		Not discussed

Difference (m₁–m₂) (s unknown)	(i) Independent samples
	(a) Large n₁, n₂—Any underlying distribution		Eq. (12.21)
	(b) Small n₁, n₂—Underlying Gaussian		Eq. (12.21)
	(c) Small n₁, n₂—Underlying non-Gaussian		Not discussed
	(ii) Paired samples	Same as for one sample after taking the difference

Relative risk	(i) Large n₁, n₂— Independent samples		Eq. (14.4)
	(ii) Large n₁, n₂—Paired samples		Same as for OR

Attributable risk	(i) Large n₁, n₂— Independent samples		Same as for p₁– p₂
	(ii) Large n₁, n₂—Paired samples		Same as for p₁– p₂

Odds ratio	(i) Large n₁, n₂— Independent samples		Eq. (14.18)
	(ii) Large n₁, n₂—Paired samples		Eq. (14.21)

RR/AR/OR	Small n		Not discussed

Summary-2 Statistical procedures for test of hypothesis on proportions

Parameter of interest and setup	Conditions	Main criterion	Equation/ Section
One dichotomous variable	Independent trials
	(a) Any n	Binomial	Use Eq. (13.1)
	(b) Large n	Gaussian Z	Eq. (13.3)

One polytomous variable	Independent trials
	(a) Large n	Goodness-of-fit chi-square	Eq. (13.5)
	(b) Small n	Multinomial	Use Eq. (13.6)

Two dichotomous variables (2´2)	(i) Two independent samples
	(a) Large n	Chi-square or Gaussian Z	Eq. (13.8) or Eq. (13.9)
	(b) Small n	Fisher’s exact	Eq. (13.11)
	(ii) Detecting a medically important difference—Large n	Gaussian Z	Eq. (13.10)
	(iii) Equivalence test	TOSTs	Sec. 13.2.2
	(iv)Matched pairs
	(a) Large n	McNemar’s	Eq. (13.12)
	(b) Small n	Binomial	Eq. (13.13)
	(v) Crossover design
	(a) Large n	Chi-square	Sec. 13.2.2
	(b) Small n	Fisher’s exact	Eq. (13.11)
Bigger tables, no matching	The case of small n not discussed in this text	Large n required

Association	2´C tables	Chi-square	Eq. (13.15)

Trend in proportions	2´C tables	Chi-square for trend	Eq. (13.16)

Association	R´C tables	Chi-square	Eq. (13.15)

Association	Three-way tables
	(i) Test of full independence	Chi-square	Eq. (13.18)
	(ii) Test of other types of independence (log-linear models)	G²	Three-way extension of Eq. (13.21)

Note: sensitivity, specificity and predictivities are proportions
Summary-3 Procedures for test of hypothesis on relative risk (RR) and odds ratio (OR)

Parameter of interest and setup	Conditions	Main criterion	Equation/ Section
Relative and attributable risks	The case of small n not discussed in this text	Large n required
ln(RR)	Two independent samples	Gaussian Z, or	Eq. (14.5)
ln(RR)	Two independent samples	Chi-square	Eq. (13.8)
RR	Matched pairs	As for OR	Sec. 14.1.2
		Gaussian Z, or McNemar’s	Eq. (14.22) or Eq. (14.23)
	Stratified	Mantel-Haenzel	Eq. (14.26)
AR	Two independent samples	Chi-square, or Gaussian Z	Eq. (13.8) or Eq. (13.9)
	Matched pairs	McNemar’s	Eq. (13.12)

Odds ratio	The case of small n not discussed in this text	Large n required
ln(OR)	Two independent samples	Chi-square	Eq. (13.8)
OR	Matched pairs	Gaussian Z, or McNemar’s	Eq. (14.22), or Eq. (14.23)
	Stratified	Mantel-Haenzel	Eq. (14.26)

Summary-4 Statistical procedures for test of hypothesis on means or locations

Setup	Conditions	Main criterion	Equations/ Sections
One sample	Comparison with prespecified—Gaussian	Student’s t	Eq. (15.1)

Comparison of two groups	(i) Paired—Gaussian	Student’s t	Eq. (15.3)
	(ii) Paired— Non-Gaussian
	(a) Any n	Sign test	Eq. (15.20a, b and c)
	(b) 5 ≤ n ≤ 19	Wilcoxon signed-ranks W_S	Eq. (15.21)
	(c) 20 ≤ n ≤ 29	Standardized W_S referred to Gaussian Z	Eq. (15.22)
	(d) n ≥ 30	Student’s t	Eq. (15.3)
	(iii) Unpaired—Gaussian	Student’s t	Eq. (15.6a) or (15.6b)
	(iv) Unpaired— Non-Gaussian
	(a) n₁, n₂ between (4, 9)	Wilcoxon rank-sum W_R	Eq. (15.23)
	(b) n₁, n₂ between (10, 29)	Standardized W_R referred to Gaussian Z	Eq. (15.24)
	(c) n₁, n₂ ≥ 30	Student’s t	Eq. (15.6a) or (15.6b)
	(v) Crossover design
	(a) Gaussian	Student’s t	Sec. 15.1.3
	(b) Non-Gaussian	Not discussed	—
	(vi) Detecting medically important difference	Student’s t	Sec. 15.4.2
	(vii) Equivalence tests	Student’s t	Sec.15.4.2

Comparison of three or more groups	(i) One-way layout
	Gaussian	ANOVA F	Eq. (15.13)
	Non-Gaussian
	(a) n ≤ 5	Kruskal-Wallis H	Eq. (15.25)
	(b) n ≥ 6	H referred to chi-square	Eq. (15.25)
	(ii) Two-way layout
	Gaussian	ANOVA F	Sec. 15.2.2
	Non-Gaussian
	(a) J ≤ 13 and K = 3	Friedman S	Eq. (15.26a) or (15.26b)
	(b) J ≤ 8 and K = 4	Friedman S	Eq. (15.26a) or (15.26b)
	(c) J ≤ 5 and K = 5	Friedman S	Eq. (15.26a) or (15.26b)
	(d) Larger J, K	S referred to chi-square	Eq. (15.26a) or (15.26b)
	(iii)Multiple comparisons
	Gaussian	Tukey D	Eq. (15.19)
	Non-Gaussian	Not discussed	—

Summary-5 Methods for studying the nature of relationship^a

Dependent variable (y)	Independent variables (xs)	Method	Section
Quantitative^b	Qualitative	ANOVA	Sec. 15.2
Quantitative	Quantitative	Quantitative regression	Chap. 16
Quantitative	Mixture of qualitative and quantitative	ANCOVA	Sec. 16.3.2

Qualitative (Dichotomous)	Qualitative or quantitative or mixture	Logistic	Sec. 17.1 and 17.2
Qualitative (Polytomous)	Qualitative or quantitative or mixture	Logistic—any two categories at a time	Sec. 17.3.2
	Qualitative	Discriminant	Sec. 19.2.3

Survival	Groups	Life table	Eq. (18.8)
		Kaplan-Meir	Eq. (18.10)
		Log-rank	Sec. 18.3.1
		Cox model	Sec. 18.3.2

^aLarge n required, particularly for tests of significance. Exact method for small n not discussed in this text.

^bQuantitative are variables on metric scale without any broad categories. Fine categories are admissible.

Summary-6 Main methods of measurement of strength of relationship between two variables

Type of variables	Measure	Equation/ Section
Both qualitative
(i) Binary categories	OR and several others	Sec. 17.5.1
(ii) Polytomous categories	Phi-coefficient	Eq. (17.7a)
	Contingency coefficient	Eq. (17.7b)
	Cramer’s V	Eq. (17.7c)
	Proportional reduction in error	Eq. (17.8)

Dependent qualitative and independent quantitative	Odds ratio	Sec. 17.1

Dependent quantitative and independent qualitative	R² from ANOVA	Eq. (17.9)

Both quantitative	R² from regression	Eq. (16.7)
(i) For linear relationship	r	Eq. (16.17)
(ii) For monotonic relationship	r_s	Eq. (16.19)
(iii) For intraclass	r_I	Eq. (16.23)

Agreement
(i) Qualitative	Cohen’s kappa	Eq. (17.10)
(ii) Quantitative	Limits of disagreement	Sec. 16.5.2
	Intraclass	Eq. (16.23)

Summary-7 Multivariate methods in different situations

Nature of the variables	Objective	Types of variables	Statistical method	Section
A dependent set and an independent set	Relationship	Dependent qualitative (independent qualitative or quantitative)	Multivariate logistic	Not discussed
	Relationship	Both quantitative	Multivariate multiple regression	Sec. 19.2.1
	Equality of means of dependents	Dependent quantitative and independent qualitative	MANOVA	Sec. 19.2.2

Dependent is one of many groups	Classify subjects into known groups	Independent quantitative	Discriminant analysis	Sec. 19.2.3
	Classify subjects into known groups	Independent qualitative or mixed	Logistic discriminant analysis	Not discussed

All variables interrelated (none is dependent)	Discover natural clusters of subjects	Qualitative or quantitative or mixed	Cluster analysis	Sec. 19.3.1
	Identify underlying factors that explain the interrelations	Quantitative	Factor analysis	Sec. 19.3.2
	Identify underlying factors that explain the interrelations	Qualitative or mixed	Factor analysis	Not discussed