Count Data Models Flashcards
(25 cards)
What do poisson/count data models focus on
- The number of occurrences of an event
- Here the outcome variable is y = 1,2,3
Give 3 examples of a poisson/count model
- Number of visits to a doctor a person makes during a year
- The number of children in a household
- The number of car accidents during a month
How do the variables work in a count model
- Samples of individuals or households will have different characteristics that affect the probability distribution
What is the probability distribution function for a poisson
- Pr(Y = y) = λ^y * e^-λ / y!
- y = 0, 1, 2, …
What does the parameter λ represent
- The expected number of times that Y = y occurs
How does λ increasing affect the distribution of the poisson
- As λ increases, the Poisson distribution shifts to the right
- For large values of λ, the Poisson distribution is approximately normal
What is the equi-dispersion property of the Poisson Distribution
- This means that the mean and variance are both equal to λ
- Equi-dispersion means that the mean is equal to variance
What is over-dispersion
- When the variance is greater than the mean
What makes a Poisson model unnatractive
- In reality, most variables are not equi-dispersed
- equi-dispersion is a strong assumption
How this the poisson regression model represented for count data
- E(Y) = λi = exp(b0 + b1 * x1i + … + bk * xki)
Why is the poisson regression model faulty
- The RHS can take any real value but the Poisson mean has to be non-negative
How do we deal with the faulty issues of the Poisson Regression model
- We replace λi with its logarthim, define ηi = ln(λi)
- The transformed mean follows a linear model ηi = ln(λi) = b0 + b1 * x1i + … + bk * xki
What does the regression coefficient represent in the logarithm count model
- Bj represents the change in the log of the λj for a uniy change in the variable xj
How else can a Poisson regression be read as
- exp(ηi) = λi = exp(b0 + b1 * x1i + … + bk * xki)
- exp(ηi) = λi = e^b0 * e^(b1 * x1i) * e^bk * xki
How is the Poisson model estimated
- Using maximum likelihood
What does the regression coefficient exp(bjh) represent when λi is modelled as as the exponential function of a linear combination
- Increasing xj by one unit multiplies λi by a factor of exp(bhj)
How do we interperet b1 for a unitary change in x, between x1 = x and x2 = x + 1
- λ1 = e^b0 * e^b1 * x
- λ2 = e^b0 * e^b1 * (x + 1)
- The ratio of two means is equal to e^b1 => λ2 / λ1 = e^b1
- Hence a change in x has a multiplicative effect on the mean of Y
What is the Incident rate ratio
- The IRR is the exponential of a Poisson regressions coefficients
- IRR = exp(β)
How can we compute the probability of observing yi
- We plug in the estimated β coefficients into the density function for a poisson
How do you calculate the marginal effect of a continuous variable
- ∂λ/∂x = ∂/∂x * exp(b0 + b1 * x) = exp(b0 + b1 * x) * (b0 + b1 * x)’ = exp(b0 + b1 * x) * b1
- The change in the conditional mean for a unit increase in x
What is the alternative way to interperet the coefficients of the model
- ln(λ) = b0 + b1 * x
- Resembles a log-linear model
- A unitary change in x leads to a 100 * b1% change in the conditional mean
How do you calculate the marginal effect of a discrete variable
- For D discrete variable, λ = exp(b0 + b1 * x1 + δ * d
- E(y|D = 1) - E(y|D = 0) = exp(b0 + b1 * x1 + δ * 1) - exp(b0 + b1 * x1 + δ * 0)
- The change in the conditional mean between the two states of D
How else can you calculate the marginal effect of a discrete variable
- ln(λ) = b0 * b1 * x + δ * D
- Going from D = 0 to D = 1 leads to a 100(e^δ - 1)% change in the conditional mean
Prove where 100 * (e^δ - 1)%
- exp(δ) = λD = 1 / λD = 0
- λD = 1 / λD = 0 - 1 = exp(δ) - 1
- λD = 1 / λD = 0 - λD = 0 / λD = 0 = exp(δ) - 1
- λD = 1 - λD = 0 / λD = 0 = exp(δ) - 1
- Then when multiplied by 100 is the percentage difference between λD = 1 and λD = 0
