BRM
BRUSHING UP
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31 min Hypothesis testing
1 h 16 Linear regression
25 min SPSS
LOGISTIC REGRESSION
watched reviewed duration topic
1 h 24 Logistic regression (I)
41 min Logistic regression (II)
40 min Logistic regression (III)
23 min Logistic regression (IV)
22 min Logistic regression (example)
23 min Logistic regression (ex.3)
Exercises: 1 – 2 – 3 – 4 – 5 – 6
FACTOR ANALYSIS
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1 h 16 Factor analysis (I)
30 min Factor analysis (II)
16 min Factor analysis (example)
19 min Factor analysis (SPSS)
Exercises: 1 – 2 – 3 – 4 – 5 –
RELIABILITY ANALYSIS
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34 min Reliability analysis
22 min Reliability analysis (SPSS)
CLUSTER ANALYSIS
watched reviewed duration topic
1h 18 Cluster analysis
10 min Cluster analysis (SPSS)
Exercises: 1 – 2 – 3 – 4 – 5
SYNTHESIS EXERCISE
watched reviewed duration topic
39 min Synthesis exercise
,Formula sheet BRM Factoranalysis
Logistic regression Factor model:
Model: 1 Xi = ai1 F1 + ai2 F2 + …+ aik Fk + Ui
P (Y = 1) = Fj = wj1 X1 + wj2 X2 + …+ wjp Xp
1 + exp(−(α + β1 x1 + β 2 x2 + ... + β p x p ))
k k
=
exp(α + β1 x1 + β 2 x2 + ... + β p x p ) ∑∑
i =1 j =1,i ≠ j
rij2
1 + exp(α + β1 x1 + β 2 x2 + ... + β p x p ) Kaiser-Meyer-Olkin : KMO = k k k k
Odds = P(Y=1) / P(Y=0) ∑∑
i =1 j =1,i ≠ j
2
ijr +∑ ∑a
i =1 j =1,i ≠ j
2
ij
Likelihood Ratio Test:
- Comparing Model 1 with Model 2, for which Model 1 is Kaiser-Meyer-Olkin for individual variable:
k
∑
nested in Model 2
- H0 : all coeff of extra variables in Model 2 are equal to 0 rij2
j =1,i ≠ j
- Test statistic = Likelihood Ratio (LR) KMOi = k k
= (-2LogLik(Model 1)) – (-2LogLik(Model 2))
- Under H0 LR has a chi-squared distribution with k degrees of ∑r
j =1,i ≠ j
2
ij + ∑a
j =1,i ≠ j
2
ij
freedom, for which k is the number of extra parameters that
are estimated in Model 2.
%&' ,
Fraction of the total variance explained by the first k (of p) factors (for
Wald test: H0: βi = 0, Wald-statistic: " = $ + ~ χ., λ1 + λ2 + L + λk
()*
PCA) =
Confidence interval for a coefficient β = b ± z* SEB p
Specificity = % of correct classifications of Y=0 2 2 2
Sensitivity = % of correct classifications of Y=1 Communality of the i-th variable = ai1 + ai 2 + L + aik
/01 23/0&2345 23/0
Standardized residual = Fraction of the total variance explained by factor j
62345 23/0 (.&2345 23/0)
Hosmer and Lemeshow test: H0: no difference between observed ( 2 2
= a1 j + a2 j + L + a pj / p
2
)
and predicted probabilities Reproduced correlation between variable i and j:
QMC when tolerance < 0.2 or VIF>5 rˆij = ai1a j1 + ai 2 a j 2 + L + aik a jk
Residual = rij − rˆij
Clusteranalysis
d ²( A, B) = ∑ ( xk , A − xk , B )
2
Squared Euclidian distance : Cronbach’s Alpha
k ∑
k
Var(itemi )
Single linkage = nearest neighbour
α= 1− i
Complete linkage = furthest neighbour
k −1 Var(scale)
UPGMA = Average linkage of between-group linkage
, BRM – Logistic regression
1. The logistic regression model
• Dependent variable Y = a binary/dichotomous/dummy variable (0/1).
• Directly estimate the probability that one of two events occur, based on a set of independent variables that can
be either quantitative or dummy variables.
• We usually estimate the probability that Y = 1.
• When you set a categorical variable with more than two categories, you have to create (n-1) dummies with one
reference group.
Why not a classical linear regression?
• Not a linear relation needed.
• Only results: 0-1.
• Thus, a classical linear regression is not optimal for a dependent variable that is binary.
General logistic regression model
• Given: Y is a binary dependent variable.
• Given: X1, X2, X…, Xp: explanatory variables, which can be either quantitative or dummy variables.
• Formula: first one on formula sheet.
Dummies
• SPSS
• (n-1) dummies with one reference category: usually first or last. Automatically done, but can be changed.
2. Regression coefficients
Estimation method
• Loglikelihood = measures how likely the observations are under the model. A high loglikelihood indicates a good
model.
• Remark: the more data, the better. You need at least 10 Y=1 (event) and Y=0 (non event) for every variable in
the model.
Interpretation in terms of probabilities
• Fill in values in formula or let SPSS do it.
• “The probability that Y=1, is estimated as …”
Interpretation in terms of odds
• Odds: ratio of the probability that an event occurs to the probability that it does not occur.
o Odds = 1: it is as likely that the event occurs (Y=1), than that it does not occur (Y=0).
o Odds < 1: it is less probable that the event occurs (Y=1), than that it does not occur (Y=0).
o Odds > 1: it is more probable that the event occurs (Y=1), than that it does not occur (Y=0).
• Odds ratio: ratio of two odds to compare two groups.
o Bi = 0 and odds ratio Exp(Bi) = 1: the ith variable has no effect on the response.
o Bi < 0 and odds ratio Exp(Bi) < 1: the ith variable has a negative effect on the response (the odds of the
event are decreased).
o Bi > 0 and odds ratio Exp(Bi) > 1: the ith variable has a positive effect on the response (the odds of the
event are increased).
• For categorical variables with more than two answers: the reference category should be used for comparison.
, Example
• Y=1 lymph nodes are cancerous and Y=0 otherwise.
• Continuous variables:
o acid
§ B = 0.024: > 0 and Exp(B) = 1.025: odds ratio > 1
§ The odds of cancerous nodes change with a factor 1.025 if acid rises with 1 unit, ceteris paribus.
Therefore, acid has a positive effect on the fact that nodes are cancerous.
o age
§ B = -0.069: < 0 and Exp(B) = 0.933: odds ratio < 1
§ The odds of cancerous nodes change with a factor 0.933 if age rises with 1 unit, ceteris paribus.
Therefore, age has a negative effect on the fact that nodes are cancerous.
• Categorical variables (dummies):
o xray
§ B = 2.045: > 0 and Exp(B) = 7.732: odds ratio > 1
§ The odds of cancerous nodes when someone has a positive x-ray result (xray = 1) is 7.732
times larger than the odds of cancerous nodes for someone who has a negative x-ray result
(xray = 0), ceteris paribus.
o stage
§ B = 1.564: > 0 and Exp(B) = 4.778: odds ratio > 1
§ The odds of cancerous nodes when someone is in an advanced stage of cancer (stage = 1) is
4.778 times larger than the odds of cancerous nodes for someone who is not in an advanced
stage of cancer (stage = 0), ceteris paribus.
o grade
§ B = 0.761: > 0 and Exp(B) = 2.1: odds ratio > 1
§ The odds of cancerous nodes when someone has an aggressively spread tumor (grade = 1) is
2.1 times larger than the odds of cancerous nodes for someone who has no aggressively
spread tumor (grade = 0), ceteris paribus.
3. Testing hypotheses about the model
Testing whether we have a useful model
• Likelihood ratio test (remember for linear regression: F-test)
• R2: be careful with the interpretation
Testing whether we have significant variables
• Wald test
• Likelihood ratio test
Likelihood ratio test: do we have a useful model? And, do we have significant variables?
• In general: full model versus reduced model.
• The higher the likelihood of a model, the better.
• The likelihood rises when extra variables are added to the model (cfr. R2 rises in linear regression). But is this
rise significant?
• Advantage: better results (vs Wald Test)
• Disadvantage: computationally more intensive (vs Wald Test)
, • The test statistic compares the likelihood of the full model with the likelihood of the reduced model:
o TS = -2 Log Lreduced – (-2 Log Lfull)
o TS ≈ Xk2 with k = difference between the number of parameters in the two models (if n large enough
and no (or very few) continuous variables in the model).
• Remark: only use models that are nested.
• Useful model?
o H0: all Bi = 0
o H1: there is a Bi ≠ 0
o We want to be able to reject H0, for a p-value < 0.05.
• The test statistic compares the likelihood of the model with the likelihood of the model that contains only the
intercept:
o TS = -2 Log Lint – (-2 Log Lmodel)
o TS ≈ Xp2
Example
• Block 0
• Block 1
• Steps:
o Block 0:
§ results of the reduced model
§ -2 Log Likelihood = 70.252
o Block 1:
§ results of the full model
§ -2 Log Likelihood = 48.126
o Test statistic:
§ Formula: -2 Log Lint – (-2 Log Lmodel) = 70.252 – 48.126 = 22.126
o Usueful model:
§ Look at Omnibus Test output: Model line: p-value = 0 < 0.05: we reject H0, meaning we have a
useful model.
§ Look at Omnibus Test output: Block line: p-value = 0 < 0.05: we reject H0, meaning at least one
variable in the model is significant
o Remark: do not delete all ‘non-significant’ variables from your model after the enter-method. It is also
interesting to see that a variable is not significant.
