List Of Numbered Equations For:

Biostatistics For The Health Sciences

Created by: R. Clifford Blair

October 4, 2008

2

Contents

2.1 The Sample Mean . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 The Population Mean . . . . . . . . . . . . . . . . . . . . . . . 82.3 The Median (n odd) . . . . . . . . . . . . . . . . . . . . . . . . 82.4 The Median (n even) . . . . . . . . . . . . . . . . . . . . . . . . 82.5 The Median (based on upper and lower limits) . . . . . . . . . . 82.6 The (exclusive) Range . . . . . . . . . . . . . . . . . . . . . . . 82.7 The (inclusive) Range . . . . . . . . . . . . . . . . . . . . . . . 82.8 Mean Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.9 The Population Variance (conceptual form) . . . . . . . . . . . 92.10 The Sample Variance (conceptual form) . . . . . . . . . . . . 92.11 The Population Variance (computational form) . . . . . . . . . 92.12 The Sample Variance (computational form) . . . . . . . . . . . 92.13 The Population Standard Deviation (conceptual form) . . . . 92.14 The Population Standard Deviation (computational form) . . . 92.15 The Sample Standard Deviation (conceptual form) . . . . . . . 92.16 The Sample Standard Deviation (computational form) . . . . . 92.17 The Percentile . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.18 The Semi-Interquartile Range . . . . . . . . . . . . . . . . . . 102.19 The Percentile Rank . . . . . . . . . . . . . . . . . . . . . . . . 102.20 The Percentile Rank (using LRL) . . . . . . . . . . . . . . . . 102.21 The Percentile Rank (using midpoint) . . . . . . . . . . . . . . 102.22 The Percentile Rank (using URL) . . . . . . . . . . . . . . . . 102.23 The Sample z Score . . . . . . . . . . . . . . . . . . . . . . . . 102.24 The Population Z Score . . . . . . . . . . . . . . . . . . . . . . 102.25 The Skew Coefficient . . . . . . . . . . . . . . . . . . . . . . . 102.26 The Kurtosis Coefficient . . . . . . . . . . . . . . . . . . . . . 113.1 The Probability of an Event . . . . . . . . . . . . . . . . . . . . 113.2 The Probability of A or B . . . . . . . . . . . . . . . . . . . . . 11

3

4 CONTENTS

3.3 The Probability of A given B . . . . . . . . . . . . . . . . . . . 113.4 A Statement of Independence (form 1) . . . . . . . . . . . . . . 113.5 A Statement of Independence (form 2) . . . . . . . . . . . . . . 113.6 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.7 Specificity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.8 Positive Predictive Value . . . . . . . . . . . . . . . . . . . . . . 113.9 Negative Predictive Value . . . . . . . . . . . . . . . . . . . . . 123.10 Prevalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.11 The Relative Risk . . . . . . . . . . . . . . . . . . . . . . . . . 123.12 The Odds Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 123.13 Bayes Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.14 Positive Predictive Value via Bayes Rule . . . . . . . . . . . . 123.15 Negative Predictive Value vis Bayes Rule . . . . . . . . . . . . 123.16 The Normal Curve . . . . . . . . . . . . . . . . . . . . . . . . 124.1 The Standard Error of The Mean . . . . . . . . . . . . . . . . . 134.2 Variance of The Mean . . . . . . . . . . . . . . . . . . . . . . . 134.3 Z Score For a Sample Mean . . . . . . . . . . . . . . . . . . . . 134.4 Standard Error of p . . . . . . . . . . . . . . . . . . . . . . . . . 134.5 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . 134.6 Z Score For Normal Approximation to The Binomial . . . . . . 134.7 Obtained Z For Test of H0 : µ = µ0 . . . . . . . . . . . . . . . . 134.8 Obtained t For Test of H0 : µ = µ0 . . . . . . . . . . . . . . . . 134.9 Obtained Z For Test of H0 : π = π0 . . . . . . . . . . . . . . . . 134.10 Zβ For Power Calculation For One Mean Z Test . . . . . . . . 144.11 Sample Size Calculation For One Mean Z Test . . . . . . . . . 144.12 Lower End of CI For µ When σ Is Known . . . . . . . . . . . . 144.13 Upper End of CI For µ When σ Is Known . . . . . . . . . . . 144.14 Lower End of CI For µ When σ Is Not Known . . . . . . . . . 144.15 Upper End of CI For µ When σ Is Not Known . . . . . . . . . 144.16 Approximate Lower End of CI For π . . . . . . . . . . . . . . . 144.17 Approximate Upper End of CI For π . . . . . . . . . . . . . . 144.18 Exact Lower End of CI For π . . . . . . . . . . . . . . . . . . . 144.19 Exact Upper End of CI For π . . . . . . . . . . . . . . . . . . 154.20 Numerator Degrees of Freedom For Lower End of Exact CI For π 154.21 Denominator Degrees of Freedom For Lower End of Exact CI

For π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.22 Numerator Degrees of Freedom For Upper End of Exact CI For π 15

CONTENTS 5

4.23 Denominator Degrees of Freedom For Upper End of Exact CIFor π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1 Paired Samples t Statistic . . . . . . . . . . . . . . . . . . . . . 155.2 Lower End of CI For µd . . . . . . . . . . . . . . . . . . . . . . 155.3 Upper End of CI For µd . . . . . . . . . . . . . . . . . . . . . . 155.4 Z Statistic For McNemar’s Test . . . . . . . . . . . . . . . . . . 155.5 χ2 Statistic For McNemar’s Test . . . . . . . . . . . . . . . . . 165.6 A Definition of The Risk Ratio . . . . . . . . . . . . . . . . . . 165.7 Sample Paired Samples Risk Ratio . . . . . . . . . . . . . . . . 165.8 Z Statistic For Testing Hypotheses Concerning RR0 . . . . . . . 165.9 Lower Limit For Paired Samples Risk Ratio . . . . . . . . . . . 165.10 Upper Limit For Paired Samples Risk Ratio . . . . . . . . . . 165.11 A Definition of The Odds Ratio . . . . . . . . . . . . . . . . . 165.12 Sample Paired Samples odds Ratio . . . . . . . . . . . . . . . 165.13 π Expressed As a Function of The Paired Samples OR . . . . . 175.14 p Expressed As a Function of The Paired Samples OR . . . . . 175.15 Lower End of Approximate CI For π . . . . . . . . . . . . . . . 175.16 Upper End of Approximate CI For π . . . . . . . . . . . . . . 175.17 Paired Samples OR Expressed As a Function of p . . . . . . . 175.18 Lower End of Approximate CI For Paired Samples OR . . . . 175.19 Upper End of Approximate CI For Paired Samples OR . . . . 175.20 Lower End of Exact Confidence For π . . . . . . . . . . . . . . 175.21 Upper End of Exact Confidence For π . . . . . . . . . . . . . . 185.22 Numerator Degrees of Freedom For Lower End of Exact CI For π 185.23 Denominator Degrees of Freedom For Lower End of Exact CI

For π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.24 Numerator Degrees of Freedom For Upper End of Exact CI For π 185.25 Denominator Degrees of Freedom For Upper End of Exact CI

For π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.1 Independent Samples t Statistic . . . . . . . . . . . . . . . . . . 186.2 Pooled Estimate of Population Variance . . . . . . . . . . . . . 186.3 Lower End of Confidence Interval For µ1 − µ2 . . . . . . . . . . 186.4 Upper End of Confidence Interval For µ1 − µ2 . . . . . . . . . . 196.5 Z Statistic For a Test of The Difference Between Proportions . 196.6 Lower End of Confidence Interval For π1 − π2 . . . . . . . . . . 196.7 Upper End of Confidence Interval For π1 − π2 . . . . . . . . . . 196.8 A Definition of The Risk Ratio . . . . . . . . . . . . . . . . . . 196.9 The Sample Independent Samples Risk Ratio . . . . . . . . . . 19

6 CONTENTS

6.10 Z Statistic For Testing Hypotheses Concerning RR0 . . . . . . 196.11 Lower End of CI For Independent Samples RR . . . . . . . . . 196.12 Upper End of CI For Independent Samples RR . . . . . . . . . 206.13 A Definition of The Odds Ratio . . . . . . . . . . . . . . . . . 206.14 The Sample Independent Samples Odds Ratio . . . . . . . . . 206.15 Z Statistic For Testing Hypotheses Concerning OR0 . . . . . . 206.16 Lower End of CI For Independent Samples OR . . . . . . . . . 206.17 Upper End of CI For Independent Samples OR . . . . . . . . . 207.1 Null Hypothesis of The Oneway ANOVA . . . . . . . . . . . . . 207.2 Oneway ANOVA F Statistic . . . . . . . . . . . . . . . . . . . . 217.3 The Mean Square Within . . . . . . . . . . . . . . . . . . . . . 217.4 The Sum of Squares Within . . . . . . . . . . . . . . . . . . . . 217.5 Computational Form of The Sum of Squares Within . . . . . . 217.6 The Mean Square Between . . . . . . . . . . . . . . . . . . . . . 217.7 The Sum of Squares Between (equal sample size version) . . . . 217.8 Sum of Squares Between . . . . . . . . . . . . . . . . . . . . . . 217.9 Null Hypothesis of 2 by k χ2 Test . . . . . . . . . . . . . . . . . 217.10 Obtained χ2 Statistic . . . . . . . . . . . . . . . . . . . . . . . 227.11 Expected Cell Frequency . . . . . . . . . . . . . . . . . . . . . 227.12 Bonferroni Adjustment . . . . . . . . . . . . . . . . . . . . . . 227.13 Tukey’s HSD Test Statistic . . . . . . . . . . . . . . . . . . . . 228.1 Pearson Product-Moment Correlation Coefficient (conceptual

form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.2 Pearson Product-Moment Correlation Coefficient (computational

form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.3 Pearson Product-Moment Correlation Coefficient (rarely used) . 228.4 t Statistic for Test of H0 : ρ = 0 . . . . . . . . . . . . . . . . . . 238.5 Statistic for Test of H0 : ρ = ρ0 . . . . . . . . . . . . . . . . . . 238.6 Lower Bound of Confidence Interval for Estimation of ρ . . . . . 238.7 Upper Bound of Confidence Interval for Estimation of ρ . . . . 238.8 Degrees of Freedom for Chi-Square Test for Independence . . . 239.1 Simple Linear Regression Model . . . . . . . . . . . . . . . . . . 239.2 Calculation of a . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.3 Calculation of b . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.4 Calculation of b When the Correlation is Known . . . . . . . . 249.5 SSy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.6 SSreg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.7 SSres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

CONTENTS 7

9.8 Coefficient of Nondetermination (1 − R2) . . . . . . . . . . . . . 249.9 Coefficient of Determination (R2) . . . . . . . . . . . . . . . . . 249.10 SSres (alternative form) . . . . . . . . . . . . . . . . . . . . . . 249.11 SSreg (alternative form) . . . . . . . . . . . . . . . . . . . . . . 249.12 Coefficient of Nondetermination (alternative form) . . . . . . . 249.13 Coefficient of Determination (alternative form) . . . . . . . . . 249.14 t Statistic For Test of H0 : β = 0 . . . . . . . . . . . . . . . . . 259.15 Lower Limit of CI For Estimation of β . . . . . . . . . . . . . 259.16 Upper Limit of CI For Estimation of β . . . . . . . . . . . . . 259.17 F Statistic For Test of H0 : R2 = 0 . . . . . . . . . . . . . . . . 259.18 Multiple Linear Regression Model . . . . . . . . . . . . . . . . 259.19 Calculation of a For Two Predictor Model . . . . . . . . . . . 259.20 Calculation of b1 For Two Predictor Model . . . . . . . . . . . 259.21 Calculation of b2 For Two Predictor Model . . . . . . . . . . . 259.22 SSreg Calculated Directly From Two Predictor Model . . . . . 259.23 F Statistic For Test of H0 : R2

y.1,···,p = 0 or H0 : β1 = β2 =· · · = βp = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9.24 F Statistic For Partial F Test . . . . . . . . . . . . . . . . . . 2610.1 Number of Permutations . . . . . . . . . . . . . . . . . . . . . 2610.2 Number of Combinations . . . . . . . . . . . . . . . . . . . . . 2610.3 Number of Ways n Subjects Can Be Assigned to k Groups . . 26

8 CONTENTS

Note: xx.xx=equation number[xxx]=page number

Chapter 2

The Sample Mean

x =

∑x

n2.1 [25]

The Population Mean

µ =

∑x

N2.2 [25]

The Median (n odd)

Median (n odd) = xn+12

2.3 [26]

The Median (n even)

Median (n even) =xn

2+ xn

2+1

22.4 [26]

The Median (based on upper and lower limits)

Median = LRL + (w)

[(.5) (n) − cf

f

]2.5 [28]

The (exclusive) Range

Range (exclusive) = xL − xS 2.6 [32]

The (inclusive) Range

Range (inclusive) = URLL − LRLS 2.7 [33]

Mean Deviation

MD =

∑|x − x|n

2.8 [34]

CONTENTS 9

The Population Variance (conceptual form)

σ2 =

∑(x − µ)2

N2.9 [35]

The Sample Variance (conceptual form)

s2 =

∑(x − x)2

n − 12.10 [35]

The Population Variance (computational form)

σ2 =

∑x2 − (

∑x)

2

N

N2.11 [36]

The Sample Variance (computational form)

s2 =

∑x2 − (

∑x)

2

n

n − 12.12 [36]

The Population Standard Deviation (conceptual form)

σ =

√√√√∑

(x − µ)2

N2.13 [37]

The Population Standard Deviation (computational form)

σ =

√√√√∑x2 − (∑

x)2

N

N2.14 [37]

The Sample Standard Deviation (conceptual form)

s =

√√√√∑

(x − x)2

n − 12.15 [37]

The Sample Standard Deviation (computational form)

s =

√√√√√∑

x2 − (∑

x)2

n

n − 12.16 [37]

10 CONTENTS

The Percentile

Pp = LRL + (w)

[(pr) (n) − cf

f

]2.17 [38]

The Semi-Interquartile Range

Q =[P75 − P25

2

]2.18 [40]

The Percentile Rank

PRP =100

[f(P−LRL)

w+ cf

]

n2.19 [40]

The Percentile Rank (using LRL)

PRP = 100

[cf

n

]2.20 [41]

The Percentile Rank (using midpoint)

PRP = 100

[(.5) (f) + cf

n

]2.21 [41]

The Percentile Rank (using URL)

PRP = 100

[f + cf

n

]2.22 [41]

The Sample z Score

z =x − x

s2.23 [42]

The Population Z Score

Z =x− µ

σ2.24 [42]

The Skew Coefficient

Skew =

∑z3

n2.25 [44]

CONTENTS 11

The Kurtosis Coefficient

Kurtosis =

∑z4

n2.26 [45]

Chapter 3

The Probability of an Event

P (A) =NA

N3.1 [52]

The Probability of A or B

P (A ∪ B) = P (A) + P (B)− P (AB) 3.2 [56]

The Probability of A given B

P (A | B) =P (AB)

P (B)3.3 [56]

A Statement of Independence (form 1)

P (A | B) = P (A) 3.4 [56]

A Statement of Independence (form 2)

P (AB) = P (A) (B) 3.5 [56]

Sensitivity

Sensitivity = P (+ | D) 3.6 [58]

Specificity

Specificity = P(− | D

)3.7 [58]

Positive Predictive Value

PPV = P (D | +) 3.8 [58]

12 CONTENTS

Negative Predictive Value

NPV = P(D | −

)3.9 [58]

Prevalence

Prevalence = P (D) 3.10 [58]

The Relative Risk

RR =P (D | E)

P(D | E

) 3.11 [59]

The Odds Ratio

OR =P (D | E)P

(D | E

)

P(D | E

)P(D | E

) 3.12 [60]

Bayes Rule

P (B | A) =P (A | B)P (B)

P (A | B)P (B) + P(A | B

)P(B) 3.13 [61]

Positive Predictive Value via Bayes Rule

PPV =(Sensitivity) (Prevalence)

(Sensitivity) (Prevalence) + (1 − Specificity) (1 − Prevalence)3.14 [61]

Negative Predictive Value vis Bayes Rule

NPV =(Specificity)(1−Prevalence)

(Specificity)(1−Prevalence)+(1−Sensitivity)(Prevalence)3.15 [62]

The Normal Curve

f (x) =1

σ√

2πe

−(x−µ)2

2σ2 3.16 [63]

Chapter 4

CONTENTS 13

The Standard Error of The Mean

σx =σ√n

4.1 [76]

Variance of The Mean

σ2x =

σ2

n4.2 [77]

Z Score For a Sample Mean

Z =x − µ

σ√

n

4.3 [78]

Standard Error of p

σp =

√π (1 − π)

n4.4 [80]

The Binomial Distribution

P (y) =n!

y! (n − y)!πy (1 − π)n−y 4.5 [81]

Z Score For Normal Approximation to The Binomial

Z =p − π√π(1−π)

n

4.6 [85]

Obtained Z For Test of H0 : µ = µ0

Z =x − µ0

σ√

n

4.7 [90]

Obtained t For Test of H0 : µ = µ0

t =x − µ0

s√

n

4.8 [102]

Obtained Z For Test of H0 : π = π0

Z =p − π0√π(1−π0)

n

4.9 [115]

14 CONTENTS

Zβ For Power Calculation For One Mean Z Test

Zβ =µ0 − µ

σx+ Zα 4.10 [132]

Sample Size Calculation For One Mean Z Test

n =σ2 (Zβ − Zα)2

(µ0 − µ)2 4.11 [136]

Lower End of CI For µ When σ Is Known

L = x − Zσ√n

4.12 [142]

Upper End of CI For µ When σ Is Known

U = x + Zσ√n

4.13 [142]

Lower End of CI For µ When σ Is Not Known

L = x − ts√n

4.14 [146]

Upper End of CI For µ When σ Is Not Known

U = x + ts√n

4.15 [146]

Approximate Lower End of CI For π

L = p − Z

√pq

n4.16 [148]

Approximate Upper End of CI For π

U = p + Z

√pq

n4.17 [148]

Exact Lower End of CI For π

L =S

S + (n − S + 1) FL

4.18 [149]

CONTENTS 15

Exact Upper End of CI For π

U =(S + 1) FU

n − S + (S + 1) FU4.19 [149]

Numerator Degrees of Freedom For Lower End of Exact CI For π

dfLN = 2 (n − S + 1) 4.20 [150]

Denominator Degrees of Freedom For Lower End of Exact CI For π

dfLD = 2S 4.21 [150]

Numerator Degrees of Freedom For Upper End of Exact CI For π

dfUN = 2 (S + 1) 4.22 [150]

Denominator Degrees of Freedom For Upper End of Exact CI For π

dfUD = 2 (n − S) 4.23 [150]

Chapter 5

Paired Samples t Statistic

t =d − µd0

sd√n

5.1 [162]

Lower End of CI For µd

L = d − tsd√n

5.2 [172]

Upper End of CI For µd

U = d + tsd√n

5.3 [172]

Z Statistic For McNemar’s Test

Z =p − .5

.50√

n

5.4 [176]

16 CONTENTS

χ2 Statistic For McNemar’s Test

χ2 =(b − c)2

b + c5.5 [176]

A Definition of The Risk Ratio

RR =P (D | E)

P(D | E

) 5.6 [190]

Sample Paired Samples Risk Ratio

RR =a + b

a + c5.7 [191]

Z Statistic For Testing Hypotheses Concerning RR0

(Useful In Equivalence Testing)

Z =ln(RR

)− ln (RR0)

√b+c

(a+b)(a+c)

5.8 [195]

Lower Limit For Paired Samples Risk Ratio

L = exp

[ln(RR

)− Z

√b + c

(a + b) (a + c)

]5.9 [196]

Upper Limit For Paired Samples Risk Ratio

U = exp

[ln(RR

)+ Z

√b + c

(a + b) (a + c)

]5.10 [197]

A Definition of The Odds Ratio

OR =P (E | D) P

(E | D

)

P(E | D

)P(E | D

) 5.11 [200]

Sample Paired Samples odds Ratio

OR =b

c5.12 [200]

CONTENTS 17

π Expressed As a Function of The Paired Samples OR

π =OR

1 + OR5.13 [205]

p Expressed As a Function of The Paired Samples OR

p =OR

1 + OR5.14 [205]

Lower End of Approximate CI For π

L = p − Z

√p (1 − p)

n5.15 [208]

Upper End of Approximate CI For π

U = p + Z

√p (1 − p)

n5.16 [208]

Paired Samples OR Expressed As a Function of p

OR =p

1 − p5.17 [208]

Lower End of Approximate CI For Paired Samples OR

L = exp

ln

(OR

)− Z

√1

b+

1

c

5.18 [209]

Upper End of Approximate CI For Paired Samples OR

U = exp

ln

(OR

)+ Z

√1

b+

1

c

5.19 [209]

Lower End of Exact Confidence For π

L =b

b + (c + 1) FL

5.20 [210]

18 CONTENTS

Upper End of Exact Confidence For π

U =(b + 1)FU

c + (b + 1) FU5.21 [210]

Numerator Degrees of Freedom For Lower End of Exact CI For π

dfLN = 2 (c + 1) 5.22 [211]

Denominator Degrees of Freedom For Lower End of Exact CI For π

dfLD = 2b 5.23 [211]

Numerator Degrees of Freedom For Upper End of Exact CI For π

dfUN = 2 (b + 1) 5.24 [211]

Denominator Degrees of Freedom For Upper End of Exact CI For π

dfUD = 2c 5.25 [211]

Chapter 6

Independent Samples t Statistic

t =x1 − x2 − δ0√s2

P

(1n1

+ 1n2

) 6.1 [219]

Pooled Estimate of Population Variance

s2p =

(∑

x21 −

(∑

x1)2

n1

)+

(∑

x22 −

(∑

x2)2

n2

)

n1 + n2 − 26.2 [220]

Lower End of Confidence Interval For µ1 − µ2

L = (x1 − x2) − t

√s2

P

(1

n1+

1

n2

)6.3 [228]

CONTENTS 19

Upper End of Confidence Interval For µ1 − µ2

U = (x1 − x2) + t

√s2

P

(1

n1+

1

n2

)6.4 [228]

Z Statistic For a Test of The Difference Between Proportions

Z =p1 − p2 − δ0√

p1q1

n1+ p2q2

n2

6.5 [232]

Lower End of Confidence Interval For π1 − π2

L = (p1 − p2) −Z

√p1q1

n1 − 1+

p2q2

n2 − 1+

1

2

(1

n1

+1

n2

) 6.6 [236]

Upper End of Confidence Interval For π1 − π2

U = (p1 − p2) +

Z

√p1q1

n1 − 1+

p2q2

n2 − 1+

1

2

(1

n1+

1

n2

) 6.7 [236]

A Definition of The Risk Ratio

RR =P (D | E)

P(D | E

) 6.8 [238]

The Sample Independent Samples Risk Ratio

RR =a/ (a + b)

c/ (c + d)6.9 [239]

Z Statistic For Testing Hypotheses Concerning RR0

Z =ln(RR

)− ln (RR0)

√b/aa+b

+ d/cc+d

6.10 [240]

Lower End of CI For Independent Samples RR

L = exp

ln

(RR

)− Z

√b/a

a + b+

d/c

c + d

6.11 [244]

20 CONTENTS

Upper End of CI For Independent Samples RR

U = exp

ln

(RR

)+ Z

√b/a

a + b+

d/c

c + d

6.12 [244]

A Definition of The Odds Ratio

OR =P (E | D) P

(E | D

)

P(E | D

)P(E | D

) 6.13 [247]

The Sample Independent Samples Odds Ratio

OR =ad

bc6.14 [249]

Z Statistic For Testing Hypotheses Concerning OR0

Z =ln(OR

)− ln (OR0)

√1a

+ 1b+ 1

c+ 1

d

6.15 [249]

Lower End of CI For Independent Samples OR

L = exp

ln

(OR

)− Z

√1

a+

1

b+

1

c+

1

d

6.16 [254]

Upper End of CI For Independent Samples OR

U = exp

ln

(OR

)+ Z

√1

a+

1

b+

1

c+

1

d

6.17 [254]

Chapter 7

Null Hypothesis of The Oneway ANOVA

H0 : µ1 = µ2 = · · · = µk 7.1 [264]

CONTENTS 21

Oneway ANOVA F Statistic

F =MSb

MSw

7.2 [264]

The Mean Square Within

MSw =SSw

N − k7.3 [265]

The Sum of Squares Within

SSw = SS1 + SS2 + · · · + SSk 7.4 [265]

Computational Form of The Sum of Squares Within

SSw =

[∑

x21 −

(∑

x1)2

n1

]+

[∑

x22 −

(∑

x2)2

n2

]+ · · · +

[∑

x2k −

(∑

xk)2

nk

]7.5 [265]

The Mean Square Between

MSb =SSb

k − 17.6 [267]

The Sum of Squares Between(equal sample size version)

SSb = n

k∑

j=1

x2j −

k∑

j=1

xj

2

k

7.7 [267]

Sum of Squares Between

SSb =

( n1∑

i=1

xi1

)2

n1+

( n2∑

i=1

xi2

)2

n2+ · · · +

( nk∑

i=1

xik

)2

nk−

(∑

All

x..

)2

N7.8 [268]

Null Hypothesis of 2 by k χ2 Test

H0 : π1 = π2 = · · · = πk 7.9 [276]

22 CONTENTS

Obtained χ2 Statistic

χ2 =∑

all cells

[(fo − fe)

2

fe

]7.10 [277]

Expected Cell Frequency

fe =(RT ) (CT )

N7.11 [279]

Bonferroni Adjustment

αPCE =αFWE

NT7.12 [285]

Tukey’s HSD Test Statistic

qij =xi − xj√

MSw

nh

7.13 [289]

Chapter 8

Pearson Product-Moment Correlation Coefficient (conceptual form)

r =

∑(x− x) (y − y)√[∑

(x − x)2] [∑

(y − y)2] 8.1 [296]

Pearson Product-Moment Correlation Coefficient (computational form)

r =

∑xy − (

∑x)(∑

y)n√√√√

[∑

x2 − (∑

x)2

n

] [∑

y2 − (∑

y)2

n

] 8.2 [296]

Pearson Product-Moment Correlation Coefficient (rarely used)

r =

∑zxzy

n − 18.3 [296]

CONTENTS 23

t Statistic for Test of H0 : ρ = 0

t =r√1−r2

n−2

8.4 [309]

Statistic for Test of H0 : ρ = ρ0

Z =.5 ln

(1+r1−r

)− .5 ln

(1+ρ0

1−ρ0

)

√1

√

n−3

8.5 [310]

Lower Bound of Confidence Interval for Estimation of ρ

L =(1 + F ) r + (1 − F )

(1 + F ) + (1 − F ) r8.6 [311]

Upper Bound of Confidence Interval for Estimation of ρ

U =(1 + F ) r − (1 − F )

(1 + F ) − (1 − F ) r8.7 [311]

Degrees of Freedom for Chi-Square Test for Independence

χ2df = (j − 1) (k − 1) 8.8 [314]

Chapter 9

Simple Linear Regression Model

y = a + bx 9.1 [320]

Calculation of a

a = y − (b) (x) 9.2 [320]

Calculation of b

b =

∑xy − (

∑x)(∑

y)n

∑x2 − (

∑x)

2

n

9.3 [320]

24 CONTENTS

Calculation of b When the Correlation is Known

b = r

√SSy

SSx

9.4 [320]

SSy

SSy = SSreg + SSres 9.5 [322]

SSreg

SSreg =∑

(y − y)2 9.6 [322]

SSres

SSres =∑

(y − y)2 9.7 [322]

Coefficient of Nondetermination (1 − R2)

1 − R2 =SSres

SSy9.8 [323]

Coefficient of Determination (R2)

R2 =SSreg

SSy9.9 [324]

SSres (alternative form)

SSres =∑

y2 − a∑

y − b∑

xy 9.10 [324]

SSreg (alternative form)

SSreg = b2SSx 9.11 [325]

Coefficient of Nondetermination (alternative form)

1 − R2 = 1 − r2yy 9.12 [326]

Coefficient of Determination (alternative form)

R2 = r2yy 9.13 [326]

CONTENTS 25

t Statistic For Test of H0 : β = 0

t =b√

MSres

SSx

9.14 [327]

Lower Limit of CI For Estimation of β

L = b − t

√MSres

SSx9.15 [327]

Upper Limit of CI For Estimation of β

U = b + t

√MSres

SSx9.16 [327]

F Statistic For Test of H0 : R2 = 0

F =R2

1−R2

n−2

9.17 [327]

Multiple Linear Regression Model

y = a + b1x1 + b2x2 + · · · + bpxp 9.18 [329]

Calculation of a For Two Predictor Model

a = y − b1x1 − b2x2 9.19 [331]

Calculation of b1 For Two Predictor Model

b1 =(SSx2) (SSyx1) − (SSx1x2) (SSyx2)

(SSx1) (SSx2) − (SSx1x2)2 9.20 [331]

Calculation of b2 For Two Predictor Model

b2 =(SSx1) (SSyx2) − (SSx1x2) (SSyx1)

(SSx1) (SSx2) − (SSx1x2)2 9.21 [331]

SSreg Calculated Directly From Two Predictor Model

SSreg = b1SSyx1 + b2SSyx2 + · · · + bpSSyxp 9.22 [331]

26 CONTENTS

F Statistic For Test of H0 : R2y.1,···,p = 0 or H0 : β1 = β2 = · · · = βp = 0

F =R2

p

1−R2

N−p−1

9.23 [332]

F Statistic For Partial F Test

F =

R2y.L

−R2y.S

pL−pS

1−R2y.L

N−pL−1

9.24 [335]

Chapter 10

Number of Permutations

Pn = n! 10.1 [345]

Number of Combinations

Cn1n2

=n!

n1!n2!10.2 [347]

Number of Ways n Subjects Can Be Assigned to k Groups

n!

n1!n2! · · ·nk!10.3 [390]