STAT 306 - Finding Relationships in Data 24-11 2026-2027 University of British Columbia

Links: Index, Third Year, Fall 2025 Term 1, STAT 306 - Finding Relationships in Data STAT 306 - Finding Relationships in Data 24-11 2026-2027 University of British Columbia #classnotes

Subject: MATH307 MATH340

#STAT306

PHYS119

Topic:: Beyond Linear Model, odds, logit, logistic regression

logistic regression - glm( response ~ covariate, family = ‘binomial’, data = my_df)

Slide 16 - Beyond Linear Model Logistic regression - predict the probability of a categorical outcome i.e. when the response variable is a categorical variable.

Show that, in general, if we increase 𝑥 by 1 unit our predicted logit will increase by 𝛽1

Show that if we increase 𝑥 by 1 unit our predicted odds is multiplied by e𝛽1

-> The division part is called the odds-ratio

Inference

For large n: if βi = 0 then ~N(0,1) ^^ test statistics - β^ − β / β ) i i SE( i For logistic regression, we always use normal distribution. For linear regression, we used tdistribution.

In R:

-> exponentiate = TRUE for odds eg. From tidy(model_cancer, conf.int = TRUE, exponentiate = TRUE)

We are 95% confident that an increase in 1 unit in perimeter is associated with an increase between 15% to 22% in the odds of cancer being malignant

Categorical Covariate

->

Categorical and Numerical Covariate

-> The male and female group have different interactions between the covariates (fare and survival). When interpreting the fare, it may different for male and female and we cannot say that we kept the sex constant.

Prediction

eg.

Slide 17 - Poisson Regression Count data: non-negative integer values that represent the number of times an event occurs within a specific time, space, or other defined interval

response variable representing count data

Poisson distribution: used for count data instead of bernoulli distribution

The rate parameter λi has to be positive

-> b Since it can be negative, we can log the rate parameter

-> when β1 is negative, the relationship is a decaying relationship otherwise its a exponential relationship

Model fitting

In R: glm(y ~ x, family = "poisson", data = df) The number of Fisher Scoring iterations will converge to the β^ , if it doesn't then it means that the design matrix X T X is not well defined ie. not full rank Interpretation the coefficients

->

when x = 0 then we have λi = eβ0

When interpreting in a response scale (non log scale - not log λi)

Inference

Wald test not in the scope of the course

z-value - Wald statistic / test statistics

In general, we want to exponentiate the CI to interpret the rate of change instead of the log rate

Deviance

Categorical covariates

The factor from a covariate k to the base covariate increases by ek The factor from a covariate k to the another covariate j increases by ek/ej Categorical and numerical covriates

For fixed xi the ratio is the same as it would be without the extra numerical covariate

Interpretation

Fix the number of months of service xi when we want to interpret the βk

Fix the type of shit βk, when we want to interpret the βxi Interaction terms -> For when we want to see if the type of ship also influences the months of service and the number of incident

Prediction