ISS 2016 Statistics Paper-2 Solution: Question 19 (General vs Generalized Linear Models)

Continuing our question-by-question walk through the ISS Statistics Paper-2 previous year papers, today we take up Question 19 from the 2016 exam. This one sits right at the boundary between the classical regression model you meet first in any statistics course and the more general framework that was built to extend it — and it is exactly the kind of "spot the difference" question the ISS examiners are fond of setting.

Quick Summary

• Topic: General Linear Model vs Generalized Linear Model

• Question Reference: ISS 2016, Statistics Paper-2, Question 19

• Correct Answer: (b) Homoscedasticity

Here is the question exactly as it appears in the paper:

19. In which one of the following assumptions do the general linear model and generalized linear model differ?

• (a) Linearity

• (b) Homoscedasticity

• (c) Unbiasedness

• (d) Nonmulticollinearity

Step 1: What Exactly Is the "General Linear Model"?

Before comparing two things, we need to be completely clear about what each one is. The general linear model is the model you almost certainly met first, usually written as:

Yi = β0 + β1X1i + β2X2i + ... + βpXpi + εi

In matrix form this is simply Y = Xβ + ε. The standard assumptions that go with this model are:

• The relationship between the mean of Y and the predictors is linear in the parameters β.

• The errors εi are normally distributed.

• The errors have constant variance σ² for every observation — this property is called homoscedasticity.

• The errors are independent of each other.

• There is no perfect linear relationship among the predictors (no perfect multicollinearity).

Notice in particular the third bullet: no matter what value X takes, the spread (variance) of Y around its mean is assumed to stay the same, σ². This is a strong assumption, and it is the one we need to keep in mind for the rest of the question.

Step 2: What Is the "Generalized Linear Model" and Why Was It Needed?

The general linear model above works beautifully when Y is roughly normal and continuous — heights, weights, exam marks, and so on. But plenty of real data does not look like that at all. Think of:

• Counts of insurance claims (Poisson-like data, always non-negative integers)

• Success/failure outcomes such as "passed" or "failed" (Binomial data, bounded between 0 and 1)

• Waiting times or amounts that are always positive and right-skewed (Gamma data)

For all of these, forcing a normal, constant-variance model is a poor fit — a count of insurance claims simply cannot have the same "spread" whether its average is 2 or 200. Nelder and Wedderburn's generalized linear model (GLM)was built precisely to handle this. A generalized linear model has three building blocks:

• Random component: Y is assumed to come from a distribution in the exponential family (Normal, Binomial, Poisson, Gamma, etc.), not necessarily Normal.

• Systematic component: a linear predictor η = Xβ, built from the explanatory variables — this part still looks exactly like the linear predictor in the general linear model.

• Link function: a function g such that g(μ) = η, connecting the mean μ of Y to the linear predictor. For Normal data with the identity link, g(μ) = μ, and the generalized linear model collapses back to the ordinary general linear model.

Step 3: Work Through Each Option One by One

Now let's take each of the four options and check, carefully, whether it is something the two models share or something that separates them.

(a) Linearity.Both models keep a linear predictor Xβ — the parameters enter the model linearly in both cases. This is exactly what makes both of them "linear" models in the first place (the "generalized linear model" name keeps the word "linear" for this very reason). So linearity is shared, not a point of difference.

(b) Homoscedasticity.In the general linear model, Var(Yi) = σ² is the same constant for every observation, regardless of the mean. In the generalized linear model, the variance is instead allowed to depend on the mean through a variance function V(μ): Var(Y) = φ·V(μ), where φ is a dispersion parameter. For example, for Poisson data V(μ) = μ, so the variance literally grows as the mean grows; for Binomial data V(μ) = μ(1 − μ). This means the constant-variance assumption is dropped in the generalized linear model — this is precisely where the two models part ways.

(c) Unbiasedness.Unbiasedness is a property of an estimator (like the least squares or maximum likelihood estimator of β), not a modeling assumption baked into either framework. Neither model "assumes" unbiasedness as such, so this option does not describe a structural difference between the two models.

(d) Nonmulticollinearity.Avoiding perfect multicollinearity among the predictors is a technical requirement for both models so that Xβ is estimable (so that XTX, or its generalized-linear-model analogue, is invertible). This requirement does not change when we move from the general linear model to the generalized linear model.

Pro Tip: The fastest way to crack this kind of "which assumption differs" question is to mentally separate a GLM into its three building blocks — random component, linear predictor, link function — and ask which block absorbs the change. Here, the linear predictor block stays identical (so linearity survives), but the random component's variance structure is what gets generalized. Students who drill themselves on this three-block breakdown until it becomes automatic rarely get tripped up by GLM questions in Paper-2, however the options are reshuffled.

Step 4: Arrive at the Final Answer

Putting Step 3 together: linearity, the need to avoid multicollinearity, and the notion of unbiasedness are common ground between the general linear model and the generalized linear model. The one genuine structural difference is that the general linear model assumes constant variance (homoscedasticity), while the generalized linear model lets the variance change with the mean through the variance function of the assumed distribution.

So the correct option is (b) Homoscedasticity. This matches the official ISS 2016 Statistics Paper-2 answer key exactly, which also lists (b) as the correct response for Question 19 — so our derivation and the official key are in full agreement.

Why This Question Matters

This question is a favourite because it tests whether you actually understand the generalized linear model as a genuine extension of the general linear model, rather than just remembering that "GLM stands for generalized linear model." Once this variance-function idea clicks, a whole family of related questions on logistic regression, Poisson regression, and link functions becomes far easier, because they are all just special cases of the same three-block structure.

Frequently Asked Questions

What is the core difference between a general linear model and a generalized linear model?

The general linear model assumes the response is Normally distributed with constant variance (homoscedasticity) and uses the identity link. The generalized linear model allows the response to follow any exponential-family distribution, connects the mean to the linear predictor through a link function, and lets the variance depend on the mean.

What is a link function, in simple terms?

A link function g is just the bridge between the mean μ of your response and the linear predictor Xβ, written as g(μ) = Xβ. For ordinary regression the link is the identity function (g(μ) = μ); for logistic regression it is the logit function, and for Poisson regression it is usually the log function.

Is the generalized linear model still a "linear" model if the response is Binomial or Poisson?

Yes. "Linear" refers to the linear predictor Xβ, not to the relationship between X and the mean of Y directly. Since the parameters β always enter the model linearly through Xβ, it is still classified as a linear model even though the link function can make the relationship between X and μ nonlinear.

Why does Poisson regression not need a homoscedasticity assumption?

Because for count data following a Poisson distribution, the variance is mathematically tied to the mean (Var(Y) = μ). Larger counts naturally have more spread than smaller counts, so assuming constant variance would misrepresent the data; the generalized linear model framework builds this mean-variance relationship in directly through the variance function.

Does unbiasedness ever come into the general linear model versus generalized linear model comparison?

Not as a defining assumption of either model — unbiasedness is a property we check for a specific estimator (such as the OLS estimator of β), and under standard conditions OLS is unbiased in the general linear model while maximum likelihood estimators in the generalized linear model are typically only asymptotically unbiased, but this is a separate discussion from the model assumptions themselves.

Where does multicollinearity fit into this comparison?

Avoiding (near) perfect multicollinearity is a practical requirement in both frameworks, because it is what allows the design matrix related computations to be inverted and the parameters β to be uniquely estimated. It is not something that changes as you move from a general linear model to a generalized linear model.

If any step above felt unclear, drop a comment below and we'll walk through it in more depth — and if you found this useful, do share it with a fellow ISS aspirant who is also working through the Paper-2 previous year questions.