Week 7 - Ratio & regression estimation

Question 1

Aim of ratio & regression estimation

Answer 1

To improve estimation of population total & pop. mean, to reduce std errors and provide less uncertainty.

How? x and y are linearly related. We use x values which are available for the whole pop (not just sample) - AUXILIARY VARIABLE of y - to PREDICT y values, which might not be available

Question 2

How to decide between using SRS vs ratio estimation vs regression estimation?

Answer 2

• use SRS estimate if no auxiliary info / low correlation / SMALL SAMPLE SIZE since likelihood of BIAS for ratio&reg. estimator
  > 1 merit is that it is UNBIASED. Both ratio & reg est.s are biased for small samples though negligible for large n.
  eg. unbiased {SRS} estimator is preferred if the ti show little variation but the Mi are very variable
• use ratio estimation if LINEAR RELATIONSHIP between x & y, with POSITIVE SLOPE through the ORIGIN
• use regression estimation if linear relationship between x & y. Slope can be -ve or +ve

> More directly, one could estimate the STANDARD ERROR for each estimator and then compare these estimates.

Question 3

How to decide whether the ratio estimator or the unbiased estimator is to be preferred? [6m]

Answer 3

Ratio estimator will be more precise and so preferred if r > 1/2 * (cv(x) / cv(y))

Question 4

How to use regression estimation & when can it provide a more precise estimation than the ratio estimation?

Answer 4

1. The regression estimation can provide more precise estimation if y is LINEARLY RELATED to x, i.e. if y = β0 + β1x and the intercept β0 ≠ 0. The scatterplot in (a) indicates that
   the straight line does not go through the origin.

Question 5

How ratio estimation may be used +
2 desirable conditions for this to be a sensible estimation method [4m, 2016]

Answer 5

Define ratio estimator in terms of variables yi (eg. fundraising amount) and xi (e.g. number of students)

Conditions:
1. relation between yi and xi is roughly LINEAR REGRESSION through ORIGIN {increasing linear relationship}
2. yi approximately PROPORTIONAL to xi // CORRELATION between yi and xi is relatively HIGH (to make the ratio estimator more precise than unbiased estimator)

Question 6

Explain why the ratio estimator may be biased. [2m, 2016]

Answer 6

Biased because the expectation of a ratio is not generally the same as the ratio of expectations

Question 7

1 example where ratio estimation works

(W7 slide 20)

Answer 7

Units = businesses
y = output measure, e.g. sales
x = measure of size, e.g. number of employees