Answer: D
“Verbal intelligence peaks in middle age” suggests a quadratic pattern: poly(Age,2)
“People with high IQs decline far less than people with low IQs” suggests an interaction between the dependence on Age and initial IQ: V0.
People decline from their teens. The rate of decline is steeper for those with a high A0. This suggests a negative coefficient on the interaction term.
The aggregated data suggests that the drug lowers recovery rate. But for each of males and females separately, the drug raises recovery rate. Since each person is either a male or a female, the correct conclusion is that the drug raises recovery rate.
Things to look for in answer:
Two different acceptable full-credit answers:
Half-credit answer:
Some background theory that I don't expect the students to give: if the study had been balanced, that is the same proportion of males and of females had been given the drug, then the “sex” vector would be orthogonal to the “drug given” vector and there would be no need for adjustment (although including “sex” would still eat up variance).
General grading principles:
Difference between a categorical and quantitative variable …
Quarter credit for each of these points:
Anything that implies these things, even it if it somewhat implicit, should be given credit.
They are welcome to say, “ordinal variables are categorical variables with a natural ordering” — if they do, take that to imply the last three points above.
“A categorical does not have a number value, e.g. sex has different levels. Quantitative is numerical.”“
Although they are wrong in saying "number value” that is a technical point. This answer makes the two points that quantitative are numerica and categorical are discrete (even if this is not stated very clearly: “sex has different levels”) 2/4 points
“Categorical – qualitative, finite range of values created to fit model. Quantitative, numerical, infinite range of values.”
2/4 points. “qualitative” is fine, as is “finite”. But “created to fit model” is meaningless. Quantitative are indeed numerical, but no reason for them to cover an infinite range.
Full credit for either of these
“The standard deviation summarizes the residuals well because as the resids get bigger, the sd also gets bigger.”
Full credit.
“Since a residual is what remains unexplained in a model, a standard deviation is a good measurement because it measures how far the case is from a typical' model value.”
1 point: “typical model value” has nothing to do with it.
“The standard deviation includes 95% of the data, discounting the outlier points and giving some indication of the size of the resids.”
0 points. Hasn't explained why the standard deviation is an appropriate measure and just wrong about discounting the outlier points.
Include an interaction term when the effect of one explanatory variable on the response depends on the level of another explanatory variable.
Give one point for saying, “Interaction terms increase \( R^2 \)”
“The interaction term gives you how different explanatory variables modulate each other in terms of their relationship to the response variable.”
4/4 points. Essential points
Take off two points for a claim that the interaction term describes how the explanatory variables influence one another; its explains the relationship to the response, not from one explanatory variable to another.
Several ways to state this:
“Because otherwise these vectors would be counted twice, making the model ambiguous and decreasing the effects of explanatory variables.”
Deserves ¾. The essential points are there (“counted twice” and “ambiguous”) but “decreasing the effects” is not on target.
“Because they can be made by having an interaction term between the other variables. Also, because redundancy brings confusion as it would be hard to tell what exact coefficients would give the best model.”
The “brings confusion” is reasonable. But “what exact coefficients would give the best model” is pretty weak. So 3 points. But then there is the “because they can be made by having an interaction term,” which is just wrong. So 2 points altogether.
Just marking the correct number will do. No need for the interpretation in terms of slopes, intercepts, and deltas.
A ~ G will produce 3 coefficients (an intercept and two deltas)A ~ B will produce 2 (a slope and an intercept)A ~ B + G will produce 4 (three parallel lines — 3 intercepts and one slope)A ~ B*G will produce 6 (three non-parallel lines — 3 intercepts and 3 slopes)Answer: A
Justification. What matters is that the density is very high near lag=0. The distribution is right-skew, so B is no good. The distribution does indeed have a long tail, but just having a long tail doesn't imply that there is high density for small lag.
Answer: D
Answer: A
April's theory is better supported by the data.
When holding nruns constant (as in model 3), the number of previous runs is positively associated with lag, not negatively, as in Fred's theory. In contrast, when holding previous constant, the eventual number of runs, nruns is negatively associated with lag.
Another way to see this is in model 4, where only data from a runner's first race is considered. Even though none of the runners have run before, there is still a negative association with the number of runs they will eventually make.
4 points for saying “April”. up 2 points more for a reasonably clear explanation.
No credit for saying “Fred” without an explanation. Up to 3 points total if they can give a compelling explanation.
Age and ethnicity
Give full credit for listing two or more plausible variables, half credit for listing just one.
Examples: education, marital status, employment, urban vs rural
Yes, there is evidence for an interaction between Age and Ethnicity. The broad pattern for each ethnicity is an increase in birth rate from teen years to the 20s or 30s, then a decrease. Asians have peak birthrate in their early 30s, while Hispanics, Blacks, and Native Americans have an earlier peak. Whites are intermediate between Asians and the other groups.
The model birth rate ~ ethnicity*age would give a straight-line pattern versus age for each ethnicity. The essential feature of birth rate for each ethnicity is that it increases with age, peaks, then decreases. So this model does is not reasonable.
birth rate ~ poly(age,2) — a second-order polynomial
No need to try to estimate coefficients. poly(age,2) or the equivalent will do entirely.
The model birthrate ~ age should get 2/6 points.
Saying “it's not a straight line” but failing to give a polynomial form should get 4/6 points.