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(Correlation and)(Multiple) Regression
Friday 5th March
(and Logistic Regression too!)
The Shape of Things to Come…Rest of Module
Week 8 Week 9 Week 10
Morning:Regression
Morning:Published MultivariateAnalyses
Morning:Log-linear
Models
Afternoon:Regression &
LogisticRegression(Computing
Session)
Afternoon:Logistic
Regression
Afternoon:Log-linear
Models(Computing
Session)
ASSESSMENT D ASSESSMENT E
The Correlation Coefficient (r)
Age at FirstChildbirth
Age at First Cohabitation
This shows the strength/closeness of a relationship
r = 0.5(or perhaps less…)
r = + 1 r = -1
r = 0
Correlation… and Regression
• r measures correlation in a linear way
• … and is connected to linear regression
• More precisely, it is r2 (r-squared) that is of relevance
• It is the ‘variation explained’ by the regression line
• … and is sometimes referred to as the ‘coefficient of determination’
y
x
Mean
The arrows show the overall variation(variation from the mean of y)
y
x
Mean
Some of the overall variation is explained by theregression line (i.e. the arrows tend to be shorter than
the dashed lines, because the regression line is closer to the points than the mean line is)
Length ofResidence (y)
Age (x)0
C
1
B
outlier
ε
y = Bx + C + ε Error term(Residual)
ConstantSlope
Regressionline
• Some variation is explained by the regression line• The residuals constitute the unexplained variation
• The regression line is chosen so as to minimise the sum of the squared residuals
• i.e. to minimise Σε2 (Σ means ‘sum of’)
• The full/specific name for this technique is
Ordinary Least Squares (OLS) linear regression
Choosing the line that best explains the data
Regression assumptions #1 and #2
0
ε
Frequency
#1: Residuals have the usual symmetric, ‘bell-shaped’ normal distribution
#2: Residuals are independent of each other
y
y
x
x
HomoscedasticitySpread of residuals (ε) stays consistent in size (range) as x increases
HomoscedasticitySpread of residuals (ε)
increases as x increases (or varies in some other way)
Use Weighted Least Squares
Regression assumption #3
Regression assumption #4
• Linearity! (We’ve already assumed this…)
• In the case of a non-linear relationship, one may be able to use a non-linear regression equation, such as:
y = B1x + B2x2 + c
Another problem: Multicollinearity
• If two ‘independent variables’, x and z, are perfectly correlated (i.e. identical), it is impossible to tell what the B values corresponding to each should be
• e.g. if y = 2x + c, and we add z, should we get:• y = 1.0x + 1.0z + c, or• y = 0.5x + 1.5z + c, or• y = -5001.0x + 5003.0z + c ?• The problem applies if two variables are highly
(but not perfectly) correlated too…
Example of Regression(from Pole and Lampard, 2002, Ch. 9)
• GHQ = (-0.69 x INCOME) + 4.94
• Is -0.69 significantly different from 0 (zero)?
• A test statistic that takes account of the ‘accuracy’ of the B of -0.69 (by dividing it by its standard error) is t = -2.142
• For this value of t in this example, the significance value is p = 0.038 < 0.05
• r-squared here is (-0.321)2 = 0.103 = 10.3%
… and of Multiple Regression
• GHQ = (-0.47 x INCOME) + (-1.95 x HOUSING) + 5.74
• For B = 0.47, t = -1.51 (& p = 0.139 > 0.05)
• For B = -1.95, t = -2.60 (& p = 0.013 < 0.05)
• The r-squared value for this regression is 0.236 (23.6%)
Interaction effects…
Squareroot oflength
of residence
Age
Women
All
Men
In this situation there is an interaction between the effects of age and of gender, so B (the slope) varies according to gender and is greater for women
Logistic regression and odds ratios
• Men: 1967/294 = 6.69 (to 1)
• Women: 1980/511 = 3.87 (to 1)
• Odds ratio 6.69/3.87 = 1.73
• Men: p/(1-p) = 3.87 x 1.73 = 6.69
• Women: p/(1-p) = 3.87 x 1 = 3.87
Odds and log odds
• Odds = Constant x Odds ratio
• Log odds = log(constant) + log(odds ratio)
• Men
log (p/(1-p)) = log(3.87) + log(1.73)
• Women
log (p/(1-p)) = log(3.87) + log(1) = log(3.87)
• log (p/(1-p)) = constant + log(odds ratio)
• Note that:
log(3.87) = 1.354
log(6.69) = 1.900
log(1.73) = 0.546
log(1) = 0
• And that the ‘reverse’ of the logarithmic transformation is exponentiation
• log (p/(1-p)) = constant + (B x SEX)
where B = log(1.73)SEX = 1 for menSEX = 0 for women
• Log odds for men = 1.354 + 0.546 = 1.900• Log odds for women
= 1.354 + 0 = 1.354
• Exp(1.900) = 6.69 & Exp(1.354) = 3.87
Interpreting effects in Logistic Regression
• In the above example: Exp(B) = Exp(log(1.73)) = 1.73 (the odds ratio!)
• In general, effects in logistic regression analysis take the form of exponentiated B’s (Exp(B)), which are odds ratios. Odds ratios have a multiplicative effect on the (odds of) the outcome
• Is a B of 0.546 (= log(1.73)) significant?• In this case p = 0.000 < 0.05 for this B.
Back from odds to probabilities
• Probability = Odds / (1 + Odds)
• Men: 6.69 / (1 + 6.69) = 0.870
• Women: 3.87 / (1 + 3.87) = 0.795
‘Multiple’ Logistic regression
• log odds = c + (B1 x SEX) + (B2 x AGE)
= c + (0.461 x SEX) + (-0.099 x AGE)
• For B1 = 0.461, p = 0.000 < 0.05
• For B2 = -0.099, p = 0.000 < 0.05
• Exp(B1) = Exp(0.461) = 1.59
• Exp(B2) = Exp(-0.099) = 0.905