List Of Numbered Equations For: Biostatistics For … · List Of Numbered Equations For:...
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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]