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Questions and Answers
What is a characteristic of linear models?
What is a characteristic of linear models?
- Changes in y are always negatively accelerated
- Changes in x produce the same change in y regardless of the value of x (correct)
- Changes in x produce different changes in y depending on the value of x
- Changes in x are always proportional to changes in y
What is an example of a scenario where a linear model might be inappropriate?
What is an example of a scenario where a linear model might be inappropriate?
- Predicting IQ from years of education
- Predicting weight from height
- Predicting height from weight
- Predicting amount of food consumed (correct)
What is a characteristic of non-linear models?
What is a characteristic of non-linear models?
- Changes in y are always positively accelerated
- Changes in x produce the same change in y regardless of the value of x
- Changes in x produce different changes in y depending on the value of x (correct)
- Changes in x are always proportional to changes in y
What is the shape of the non-linear model described in the lecture if changes in y get lower as x increases?
What is the shape of the non-linear model described in the lecture if changes in y get lower as x increases?
What is the primary reason for using the generalized linear model?
What is the primary reason for using the generalized linear model?
Which link function is used for modeling counts or frequencies?
Which link function is used for modeling counts or frequencies?
What is the range of the predictors in logistic regression and what does the logistic function ensure about those predictors
What is the range of the predictors in logistic regression and what does the logistic function ensure about those predictors
What is the primary advantage of using the logistic function in logistic regression?
What is the primary advantage of using the logistic function in logistic regression?
What is the link used as a function of Y in the Genearlised Linear Model for binary variables?
What is the link used as a function of Y in the Genearlised Linear Model for binary variables?
What is the purpose of the link function in the generalized linear model?
What is the purpose of the link function in the generalized linear model?
What is the main advantage of binary data not having an assumption of normality?
What is the main advantage of binary data not having an assumption of normality?
There is no homoscedasticity for logistic regression (equal variance) because
There is no homoscedasticity for logistic regression (equal variance) because
What is the method used to fit the model to the data in binary logistic regression?
What is the method used to fit the model to the data in binary logistic regression?
What is the distribution of binary outcomes in binary logistic regression?
What is the distribution of binary outcomes in binary logistic regression?
What is the assumption about the observations in binary logistic regression?
What is the assumption about the observations in binary logistic regression?
What is the result of exponentiating the log odds of 0.190?
What is the result of exponentiating the log odds of 0.190?
What is the relationship between the predictors and the probability in a logistic model?
What is the relationship between the predictors and the probability in a logistic model?
What is the effect of a one-unit increase in parental education on the log odds of taking algebra 2?
What is the effect of a one-unit increase in parental education on the log odds of taking algebra 2?
What is the relationship between the odds and the predictors in a logistic model?
What is the relationship between the odds and the predictors in a logistic model?
What is the purpose of exponentiating log odds?
What is the purpose of exponentiating log odds?
What is the primary method used to estimate parameters in logistic regression?
What is the primary method used to estimate parameters in logistic regression?
Why is it easier to account for more variance in logistic regression when the mean is close to 0 or 1?
Why is it easier to account for more variance in logistic regression when the mean is close to 0 or 1?
What is the basis for calculating R2 in logistic regression?
What is the basis for calculating R2 in logistic regression?
What adjustment is made to the Cox and Snell R2 by Nagelkerke?
What adjustment is made to the Cox and Snell R2 by Nagelkerke?
Why is it not possible to compare the variance explained in logistic regression to that of linear regression?
Why is it not possible to compare the variance explained in logistic regression to that of linear regression?
What does it mean to correctly predict values in logistic regression?
What does it mean to correctly predict values in logistic regression?
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Study Notes
Binary Data and Logistic Regression
- Binary data come from a binomial or Bernoulli distribution
- There are no assumptions of normality, linearity, or homoscedasticity (equal variances)
- The variance of binomially distributed data depends on the probability or frequency
- Variability is highest for intermediate probabilities and decreases for extreme probabilities (p = 0.02 and 0.99)
Assumptions of Logistic Regression
- Binary outcomes that are mutually exclusive
- Independence of observations (as per usual)
- Independent variables can be continuous or categorical
- Model is fit to data using maximum likelihood estimation
How Logistic Regression Works
- Predicted probabilities are calculated from the model for each observation (0 or 1)
- Likelihood of each observation is calculated according to a Bernoulli distribution
- The closer the predicted probabilities are to the data, the higher the likelihood
Linear vs Non-Linear Models
- Linear models: changes in x produce the same change in y regardless of the value of x
- Non-linear models: changes in x produce changes in y that depend on the value of x
- Logistic regression is a non-linear model that predicts a binary variable from other variables
Generalized Linear Model
- A way to explore datasets that do not conform to the assumptions of linear regression
- The appropriate function/link allows linear techniques to be employed, even if the data are not linear
Logistic Regression Equation
- Regression equation is substituted into the logistic function
- Predictors can range from negative infinity to infinity, but the logistic function makes it such that the predictors only predict values between 0 and 1
Interpreting Logistic Regression Results
- A value of 1 is predicted if the predicted probability is > 0.5, and a value of 0 is predicted if the predicted probability is < 0.5
- B is the log odds ratio
- Log odds increase by a certain value for every unit increase in the predictor
- Odds are exponentially related to the predictors, and probability is linked to the predictors via a logistic function
Model Evaluation
- Models should be compared using the Cox & Snell/Nagelkerke R2
- R2 is not calculated based on the correlation or variation accounted for, but rather on likelihood ratios
- The Cox and Snell (1989) approximation is a ratio of the log likelihood of the data under the model to the log likelihood of the data under the "null model" (no predictors)
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