Post on 12-Jan-2016
description
N-way ANOVA
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3-way ANOVA
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3-way ANOVA
H0: The mean respiratory rate is the same for all speciesH0: The mean respiratory rate is the same for all temperaturesH0: The mean respiratory rate is the same for both sexes H0: The mean respiratory rate is the same for all speciesH0: There is no interaction between species and temperature across
both sexesH0: There is no interaction between species and sexes across
temperatureH0: There is no interaction between sexes and temperature across
both spicesH0: There is no interaction between species, temperature, and sexes
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3-way ANOVA Latin Square
Multiple and non-linear regression
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What is what?
• Regression: One variable is considered dependent on the other(s)• Correlation: No variables are considered dependent on the other(s)• Multiple regression: More than one independent variable• Linear regression: The independent factor is scalar and linearly
dependent on the independent factor(s)• Logistic regression: The independent factor is categorical
(hopefully only two levels) and follows a s-shaped relation.
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Remember the simple linear regression?
If Y is linaery dependent on X, simple linear regression is used:
is the intercept, the value of Y when X = 0
is the slope, the rate in which Y increases when X increases
jj XY
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I the relation linaer?
-3 -2 -1 0 1 2 3-4
-2
0
2
4
6
8
10
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Multiple linear regression
If Y is linaery dependent on more than one independent variable:
is the intercept, the value of Y when X1 and X2 = 01 and 2 are termed partial regression coefficients1 expresses the change of Y for one unit of X when 2 is kept constant
jjj XXY 2211
05
1015
20
25
1
2
3
4
5
6
70
0.5
1
1.5
2
2.5
3
3.5
4
4.5
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Multiple linear regression – residual error and estimations
As the collected data is not expected to fall in a plane an error term must be added
The error term summes up to be zero.
Estimating the dependent factor and the population parameters:
jjjj XXY 2211
05
1015
20
25
1
2
3
4
5
6
70
0.5
1
1.5
2
2.5
3
3.5
4
4.5
jjj XbXbaY 2211ˆ
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Multiple linear regression – general equations
In general an finite number (m) of independent variables may be used to estimate the hyperplane
The number of sample points must be two more than the number of variables
j
m
iijij XY
1
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Multiple linear regression – least sum of squares
The principle of the least sum of squares are usually used to perform the fit:
2
1
ˆ
n
jjj YY
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Multiple linear regression – An example
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Multiple linear regression – The fitted equation
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Multiple linear regression – Are any of the coefficients significant?
F = regression MS / residual MS
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Multiple linear regression – Is it a good fit?
R2 = 1-regression SS / total SS• Is an expression of how much of
the variation can be described by the model
• When comparing models with different numbers of variables the ajusted R-square should be used:
Ra2 = 1 – regression MS / total MS
The multiple regression coefficient:R = sqrt(R2) The standard error of the estimate =
sqrt(residual MS)
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Multiple linear regression – Which of the coefficient are significant?
• sbi is the standard error of the regresion parameter bi
• t-test tests if bi is different from 0
• t = bi / sbi
• is the residual DF• p values can be found in a
table
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Multiple linear regression – Which of the are most important?
• The standardized regression coefficient , b’ is a normalized version of b
2
2'
y
xbb iii
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Multiple linear regression - multicollinearity
• If two factors are well correlated the estimated b’s becomes inaccurate.
• Collinearity, intercorrelation, nonorthogonality, illconditioning• Tolerance or variance inflation factors can be computed
• Extreme correlation is called singularity and on of the correlated variables must be removed.
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Multiple linear regression – Pairvise correlation coefficients
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22;;
XXxYYXXxy
yx
xyr iiixy
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Multiple linear regression – Assumptions
The same as for simple linear regression:1. Y’s are randomly sampled 2. The reciduals are normal distributed 3. The reciduals hav equal variance4. The X’s are fixed factors (their error are small). 5. The X’s are not perfectly correlated
Logistic regression
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Logistic Regression
• If the dependent variable is categorical and especially binary?
• Use some interpolation method
• Linear regression cannot help us.
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The sigmodal curve
0 1 1
1
1 e...
z
n n
p
z x x
-6 -4 -2 0 2 4 60
0.2
0.4
0.6
0.8
1
x
p
sigmodal curve
0 = 0;
1 = 1
XX
X
ee
ep
1
1
1
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The sigmodal curve
• The intercept basically just ‘scale’ the input variable
0 1 1
1
1 e...
z
n n
p
z x x
-6 -4 -2 0 2 4 60
0.2
0.4
0.6
0.8
1
x
p
sigmodal curve
0 = 0;
1 = 1
0 = 2;
1 = 1
0 = -2;
1 = 1
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The sigmodal curve
0 1 1
1
1 e...
z
n n
p
z x x
-6 -4 -2 0 2 4 60
0.2
0.4
0.6
0.8
1
x
p
sigmodal curve
0 = 0;
1 = 1
0 = 0;
1 = 2
0 = 0;
1 = 0.5
• The intercept basically just ‘scale’ the input variable
• Large regression coefficient → risk factor strongly influences the probability
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The sigmodal curve
0 1 1
1
1 e...
z
n n
p
z x x
-6 -4 -2 0 2 4 60
0.2
0.4
0.6
0.8
1
x
p
sigmodal curve
0 = 0;
1 = 1
0 = 0;
1 = -1
• The intercept basically just ‘scale’ the input variable
• Large regression coefficient → risk factor strongly influences the probability
• Positive regression coefficient → risk factor increases the probability
• Logistic regession uses maximum likelihood estimation, not least square estimation
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Does age influence the diagnosis? Continuous independent variable
Variables in the Equation
B S.E. Wald df Sig. Exp(B)
95% C.I.for EXP(B)
Lower Upper
Step 1a Age ,109 ,010 108,745 1 ,000 1,115 1,092 1,138
Constant -4,213 ,423 99,097 1 ,000 ,015
a. Variable(s) entered on step 1: Age.
age1
1
10
BBze
pz
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Does previous intake of OCP influence the diagnosis? Categorical independent variable
Variables in the Equation
B S.E. Wald df Sig. Exp(B)
95% C.I.for EXP(B)
Lower Upper
Step 1a OCP(1) -,311 ,180 2,979 1 ,084 ,733 ,515 1,043
Constant ,233 ,123 3,583 1 ,058 1,263
a. Variable(s) entered on step 1: OCP.
OCP1
1
10
BBze
pz
0.48051
1
1
1)1( 1, OCP If
0.55801
1
1
1)1( 0, OCP If
311.0233.01
233.0
10
0
eeYp
eeYp
BB
B
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Odds ratio
zep
po
1
0.7327 ratio odds 311.01010
0
10
eeee
e BBBBB
BB
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Multiple logistic regression
Variables in the Equation
B S.E. Wald df Sig. Exp(B)
95% C.I.for EXP(B)
Lower Upper
Step 1a Age ,123 ,011 115,343 1 ,000 1,131 1,106 1,157
BMI ,083 ,019 18,732 1 ,000 1,087 1,046 1,128
OCP ,528 ,219 5,808 1 ,016 1,695 1,104 2,603
Constant -6,974 ,762 83,777 1 ,000 ,001
a. Variable(s) entered on step 1: Age, BMI, OCP.
BMIageOCP1
1
3210
BBBBze
pz
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Predicting the diagnosis by logistic regression
What is the probability that the tumor of a 50 year old woman who has been using OCP and has a BMI of 26 is malignant?
z = -6.974 + 0.123*50 + 0.083*26 + 0.28*1 = 1.6140p = 1/(1+e-1.6140) = 0.8340
Variables in the Equation
B S.E. Wald df Sig. Exp(B)
95% C.I.for EXP(B)
Lower Upper
Step 1a Age ,123 ,011 115,343 1 ,000 1,131 1,106 1,157
BMI ,083 ,019 18,732 1 ,000 1,087 1,046 1,128
OCP ,528 ,219 5,808 1 ,016 1,695 1,104 2,603
Constant -6,974 ,762 83,777 1 ,000 ,001
a. Variable(s) entered on step 1: Age, BMI, OCP.
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Exercises
20.1, 20.2
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Exercises
14.1, 14.2