Logistic Regression correct answers Commonly used for modeling binary response data. The response variable is a binary variable, and thus, not normally distributed.
In logistic regression, we model the probability of a success, not the response variable. In this model, we do not have an error t...
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Logistic Regression correct answers Commonly used for modeling binary response data. The
response variable is a binary variable, and thus, not normally distributed.
In logistic regression, we model the probability of a success, not the response variable. In this
model, we do not have an error term
g-function correct answers We link the probability of success to the predicting variables using
the g link function. The g function is the s-shape function that models the probability of success
with respect to the predicting variables
The link function g is the log of the ratio of p over one minus p, where p again is the probability
of success
Logit function (log odds function) of the probability of success is a linear model in the predicting
variables
The probability of success is equal to the ratio between the exponential of the linear combination
of the predicting variables over 1 plus this same exponential
Odds of a success correct answers This is the exponential of the Logit function
Logistic Regression Assumptions correct answers Linearity: The relationship between the g of
the probability of success and the predicted variable, is a linear function.
Independence: The response binary variables are independently observed
Logit: The logistic regression model assumes that the link function g is a logit function
Linearity Assumption correct answers The Logit transformation of the probability of success is a
linear combination of the predicting variables. The relationship may not be linear, however, and
transformation may improve the fit
The linearity assumption can be evaluated by plotting the logit of the success rate versus the
predicting variables.
If there's a curvature or some non-linear pattern, it may be an indication that the lack of fit may
be due to the non-linearity with respect to some of the predicting variables
Logistic Regression Coefficient correct answers We interpret the regression coefficient beta as
the log of the odds ratio for an increase of one unit in the predicting variable
We do not interpret beta with respect to the response variable but with respect to the odds of
success
,The estimators for the regression coefficients in logistic regression are unbiased and thus the
mean of the approximate normal distribution is beta. The variance of the estimator does not have
a closed form expression
Model parameters correct answers The model parameters are the regression coefficients.
There is no additional parameter to model the variance since there's no error term.
For P predictors, we have P + 1 regression coefficients for a model with intercept (beta 0).
We estimate the model parameters using the maximum likelihood estimation approach
Response variable correct answers The response data are Bernoulli or binomial with one trial
with probability of success
MLE correct answers The resulting log-likelihood function to be maximized, is very complicated
and it is non-linear in the regression coefficients beta 0, beta 1, and beta p
MLE has good statistical properties under the assumption of a large sample size i.e. large N
For large N, the sampling distribution of MLEs can be approximated by a normal distribution
The least square estimation for the standard regression model is equivalent with MLE, under the
assumption of normality.
MLE is the most applied estimation approach
Parameter estimation correct answers Maximizing the log likelihood function with respect to
beta0, beta1 etc in closed (exact) form expression is not possible because the log likelihood
function is a non-linear function in the model parameters i.e. we cannot derive the estimated
regression coefficients in an exact form
Use numerical algorithm to estimate betas (maximize the log likelihood function). The estimated
parameters and their standard errors are approximate estimates
Binomial Data correct answers This is binary data with repititions
Marginal Relationship correct answers Capturing the association of a predicting variable to the
response variable without consideration of other factors
Conditional Relationship correct answers Capturing the association oof a predicting variable to
the response variable conditional of other predicting variables in the model
Simpson's paradox correct answers This is when the addition of a predictive variable reverses the
sign on the coefficients of an existing parameter
, It refers to reversal of an association when looking at a marginal relationship versus a partial or
conditional one. This is a situation where the marginal relationship adds a wrong sign
This happens when the 2 variables are correlated
Normal Distribution correct answers Normal distribution relies on a large sample of data. Using
this approximate normal distribution we can further derive confidence intervals.
Since the distribution is normal, the confidence interval is the z-interval
**Applies for Logistic & Poisson Regression
Hypothesis Testing (coefficient == 0) correct answers To perform hypothesis testing, we can use
the approximate normal sampling distribution.
The resulting hypothesis test is also called the Wald test since it relies on the large sample
normal approximation of MLEs
To test whether the coefficient betaj = 0 or not, we can use the z- value
**Applies for Logistic & Poisson Regression
Wald Test (Z-test) correct answers The z-test value is the ratio between the estimated coefficient
minus 0, (which is the null value) divided by the standard error
We reject the null hypothesis that the regression coefficient is 0 if the z value (gets too large) is
larger in absolute value than the z critical point, (or the 1- alpha over 2 of the normal quantile).
We interpret that the coefficient is statistically significant
**Applies for Logistic & Poisson Regression
Hypothesis Testing (coefficient == constant) correct answers To test if the regression coefficient
is equal to this constant b, then the z-value changes.
We subtract b from the estimated coefficients of the numerator
We decide to reject/accept using the P-value
The P-value is 2 times the left tail of the standard normal of the quantile provided by the absolute
value of the z-value
P-value = 2P(Z > |z-value|)
**Applies for Logistic & Poisson Regression
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