Lecture 8

Inference for Complicated Summaries

Introduction

$0k $25k $50k $75k $100k $125k $150k $175k $200k 8y 9y 10y 11y 12y 13y 14y 15y 16y 17y 18y 19y 20y

$0k $25k $50k $75k $100k $125k $150k $175k $200k 8y 9y 10y 11y 12y 13y 14y 15y 16y 17y 18y 19y 20y

Population

\[ \small{ \begin{array}{ccccccc} x_1 & y_1 & x_2 & y_2 & \ldots & x_m & y_m \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \end{array} } \]

Sample

\[ \small{ \begin{array}{ccccccc} X_1 & Y_1 & X_2 & Y_2 & \ldots & X_n & Y_n \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \end{array} } \]

  • Last class, we talked about summaries of the relationship between income and education in a population.
  • Today, we’ll talk about estimating them by summarizing a sample drawn from it exactly the same way.
  • This is a recipe we’ve been using since the start of the semester.
    • We’ve used sample means estimate population means. \[ \frac1n\sum_{i=1}^n Y_i \qqtext{ estimates } \frac1m\sum_{j=1}^m y_j \]
    • We’ve used subsample means to estimate subpopulation means. \[ \frac{1}{N_x}\sum_{i:X_i=x}Y_i \qqtext{ estimates } \frac{1}{m_x}\sum_{j:x_j=x}y_j \qfor N_x=\sum_{i:X_i=x} 1 \qand m_x = \sum_{j:x_j=x}1 \]
  • What’ll be new is that the summaries we’ll be estimating will be more complex.
    • For many of them, we already have the tools to understand estimation and inference.
    • For others, we’ll need new tools that we’ll cover today.
    • Let’s review a few examples.

$0k $25k $50k $75k $100k $125k $150k $175k $200k 8y 9y 10y 11y 12y 13y 14y 15y 16y 17y 18y 19y 20y

\[ \hat\theta = \hat\mu(16) - \hat\mu(12) \]

$0k $25k $50k $75k $100k $125k $150k $175k $200k 8y 9y 10y 11y 12y 13y 14y 15y 16y 17y 18y 19y 20y

\[ \begin{aligned} \hat\theta &= \hat\mu(16) - \hat\mu(\le 12) && \qfor \hat\mu(\le 12) = \frac{\sum_{i:X_i \le 12} \hat\mu(X_i)}{\sum_{i:X_i \le 12} 1} \end{aligned} \]

  • Same thing. We can ‘forget’ that we have the resolution we don’t want.
  • If our data had come to us dichotomized already, the tools we have would apply.
  • The math doesn’t care who dichotomized the data, so we can use the same tools.

$0k $25k $50k $75k $100k $125k $150k $175k $200k 8y 9y 10y 11y 12y 13y 14y 15y 16y 17y 18y 19y 20y

\[ \begin{aligned} \hat\theta &= \frac{\sum\limits_{x=9}^{12} \qty{ \hat\mu(x) - \hat\mu(x-1) }}{4} && \qqtext{ the average over years} \\ \hat\theta &= \frac{\sum\limits_{i:X_i \in 9 \ldots 12} \qty{ \hat\mu(X_i) - \hat\mu(X_i-1) }}{\sum_\limits{i:X_i \in 9 \ldots 12} 1} && \qqtext{ the average over people} \end{aligned} \]

  • These are different. There’s no way to ‘forget’ into a situation we’ve already addressed.
  • We’ll need some new tools to deal with these. That’s what we’ll work on today.

Quiz

$0k $25k $50k 8y 9y 10y 11y 12y

\[ \begin{aligned} \hat\theta_A &= \frac{\sum\limits_{x=9}^{12} \qty{ \hat\mu(x) - \hat\mu(x-1) }}{4} && \qqtext{ the average over years} \\ \hat\theta_B &= \frac{\sum\limits_{i:X_i \in 9 \ldots 12} \qty{ \hat\mu(X_i) - \hat\mu(X_i-1) }}{\sum_\limits{i:X_i \in 9 \ldots 12} 1} && \qqtext{ the average over people} \end{aligned} \]

  1. One of these two estimators can be ‘unrolled’ into a simple comparison of two column means. \[ \textcolor[RGB]{17,138,178}{\hat \theta = \frac{\hat \mu(12) - \hat \mu(8)}{4}} \] Which is it?

  2. What is the variance of this estimator \(\textcolor[RGB]{17,138,178}{\hat\theta}\)?

  • You may express your answer in terms of anything that appears in previous lectures.
  • e.g., \(\mathop{\mathrm{\mathop{\mathrm{V}}}}[\hat \mu(7)]\) can appear in your answer if you want it to.

Quiz Solution

$0k $25k $50k 8y 9y 10y 11y 12y

It’s the average over years that unrolls like this. \[ \begin{aligned} \frac{\sum_{x=9}^{12} \qty{ \hat\mu(x) - \hat\mu(x-1) }}{4} &= \frac{ \qty{\mu(12) - \mu(11)} + \qty{ \mu(11) - \mu(10) } + \qty{ \mu(10) - \mu(9) } + \qty{ \mu(9) - \mu(8) } }{4} \\ &= \frac{\mu(12) - \mu(8)}{4} \end{aligned} \]

It’s \(1/4^2 = 1/16\) times the variance of the difference of these two column means.

  • For any random variable \(X\) and constant \(a\) … \[ \mathop{\mathrm{\mathop{\mathrm{V}}}}[aX] \overset{\texttip{\text{ \ ❓ \ }}{definition}}{=} \mathop{\mathrm{E}}[(aX-\mathop{\mathrm{E}}[aX])^2] \overset{\texttip{\text{ \ ❓ \ }}{linearity of expectation + arithmetic}}{=} a^2\mathop{\mathrm{E}}[(X-\mathop{\mathrm{E}}[X])^2] \overset{\texttip{\text{ \ ❓ \ }}{definition}}{=} a^2\mathop{\mathrm{\mathop{\mathrm{V}}}}[X]. \]

  • In the special case \(a=1/4\) and \(X=\hat\mu(12) - \hat\mu(8)\) … \[ \begin{aligned} \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[\frac{\hat\mu(12) - \hat\mu(8)}{4}] &= \frac{1}{4^2}\mathop{\mathrm{\mathop{\mathrm{V}}}}[\hat\mu(12) - \hat\mu(8)] && \text{ is the answer I was looking for } \\ &= \frac{1}{16} \times \mathop{\mathrm{E}}\qty[ \frac{\sigma^2(12)}{N_12} + \frac{\sigma^2(8)}{N_8} ] && \text{ using a formula from our lecture on comparing two groups.} \end{aligned} \]

We know this estimator is unbiased. We can think about the difference in means for this, too. \[ \begin{aligned} \mathop{\mathrm{E}}\qty[\frac{\hat\mu(12) - \hat\mu(8)}{4}] &\overset{\texttip{\text{ \ ❓ \ }}{via linearity}}{=} \frac{\mathop{\mathrm{E}}[\hat\mu(12)] - \mathop{\mathrm{E}}[\hat\mu(8)]]}{4} \\ &\overset{\texttip{\text{ \ ❓ \ }}{via unbiasedness of subsample means}}{=} \frac{\mu(12) - \mu(8)}{4}\\ &\overset{\texttip{\text{ \ ❓ \ }}{'rolling it back up'}}{=} \frac{1}{4}\sum_{x=9}^{12}\qty{\mu(x) - \mu(x-1)}{4}. \end{aligned} \]

What This Tells Us About Inference

0e+00 1e-04 2e-04
Our estimator’s actual sampling distribution
0e+00 1e-04 2e-04 -651.1175 2369.3952 5389.9080
Our estimate of that sampling distribution

To turn a point estimator into a confidence interval, we need to know two things about its sampling distribution.

  • Where it’s centered—relative to the estimation target.
    • In this case, it’s exactly were we want it to be.
    • Our estimator is unbiased, so it’s centered at the target.
  • How wide it is, i.e. how far out from center we need to go to cover 95% of draws. ●s
    • For this, we can use an estimate of the sampling distribution.
    • Using one based on normal approximation, that’s \(\pm 1.96 \ \text{estimated standard deviations}\)
  • To illustrate what’s going on, we can draw the picture on the left above. We …
    • Place our estimated sampling distribution right at the target. It’ll have roughly the same shape.
    • Think of our estimate as one of many draws ● from it and draw arms on each.
    • See that, for 95% of these draws, the arms touch the target at the distribution’s center.
  • We don’t know where this center \(\theta\) is. Otherwise we wouldn’t bother estimating it.
    • That’s why I haven’t labeled the ticks on the \(x\)-axis.
    • But we don’t need to know that to know our intervals are calibrated.
    • Calibration is about where our draws are relative to our estimation target. Unbiasedness is enough.
  • We can understand and anticipate the width of these intervals using our formula for the estimator’s variance.
  • All we need are estimates of a few things in the formula: subpopulation standard deviations and subsample sizes.
  • We can use this knowledge to design studies that give us a desired level of precision.
    • We can use a pilot study to estimate these things.
    • And use those estimates to choose an appropriate sample size for a second wave.
    • This is what we imagined doing when we talked about the NSW Demonstration a few weeks back.

\[ \begin{aligned} \mathop{\mathrm{\mathop{\mathrm{V}}}}[\hat\theta] &= \frac{1}{16} \times \mathop{\mathrm{E}}\qty[ \frac{\sigma^2(12)}{N_{12}} + \frac{\sigma^2(8)} {N_8} ] \\ &\approx \frac{1}{16} \times \qty[\frac{\hat\sigma^2(12)}{N_{12}} + \frac{\hat\sigma^2(8)}{N_8} ] \end{aligned} \]

\(x\) \(N_x\) \(\hat \mu(x)\) \(\hat \sigma(x)\)
8 14 19.3K 22.2K
12 605 28.8K 28.3K

Calibration for the Average over People

0e+00 1e-04 2e-04 3e-04 0 5000 10000 15000
Bootstrap Sampling Distribution and Interval Estimate for the Average over Years
0e+00 1e-04 2e-04 3e-04 0 5000 10000 15000
Bootstrap Sampling Distribution and Interval Estimate for the Average over People
  • We don’t yet know these things about our other summary: the average over people.
  • Our first step will be to show that it, too, is unbiased.
    • That’s really enough for calibration.
    • We can use the bootstrap sampling distribution to get the right width.
  • Our second step will be to work out a formula for its variance.
    • This’ll allow us to understand our estimator better, design properly sized-studies, etc.
    • And it’ll be the key to understanding a phenomenon we can see in the plots above.
  • The average over people is harder to estimate than the average over years. At least using this data.
    • We can see its bootstrap sampling distribution is at least twice as wide.
    • What’s behind this? We’ll find out.

Unbiasedness

Warm-Up

$0k $25k $50k 8y 9y 10y 11y 12y

  • When analyzing estimators like this, it’s helpful to rewrite them as a linear combination of column means.
  • That’s what we were doing when we ‘unrolled’ our average over years to see that it was the secant slope.
  • And it’s helpful, if possible, to work out formulas for its coefficients with as few cases as possible.
    • Why? Because when we do calculations, we have to analyze each case.
    • In this example, 3 cases is enough. It often is.
  • Give it a shot. Fill in the cases below.

\[ \begin{aligned} \frac{1}{4}\sum_{x\in 9 \ldots 12} \qty{ \hat \mu(x) - \hat \mu(x-1)} &= \frac{\hat\mu(12) -\hat\mu(8)}{4} &= \textcolor[RGB]{17,138,178}{\sum_{x}\hat\alpha(x)\hat\mu(x)} \qfor \hat\alpha(x) = \begin{cases} & \qqtext{if} \\ & \qqtext{if} \\ & \qqtext{if} \\ \end{cases} \end{aligned} \]

Step 1. Rewriting the Average over People in Standard Form

\[ \frac{\sum_{i:X_i \in 9 \ldots 12} \qty{ \hat \mu(X_i) - \hat \mu(X_i-1)}}{\sum_{i:X_i \in 9 \ldots 12} 1} = \sum_{x}\hat\alpha(x)\hat\mu(x) \qfor \hat\alpha(x)= \]

  • We’ll do this in three steps.
    • We’ll rewrite it as a linear combination of differences in subsample means.
      • This’ll be a lot like what we did for aggregate means last class.
    • We’ll ‘unroll’ that sum and group terms multiplying each subsample mean.
      • We did this for the average over years and got something very simple.
      • All but two coefficients \(\hat\alpha(x)\) were zero.
      • This one will be messier.
    • We’ll try to recognize a pattern in the coefficients so we don’t have to analyze every term by itself.

\[ \begin{aligned} \frac{\sum_{i:X_i \in 9 \ldots 12} \qty{ \hat\mu(X_i) - \hat\mu(X_i-1) }}{\sum_{i:X_i \in 9 \ldots 12} 1} &= \sum_{x \in 9 \ldots 12} P_x \ \qty{ \hat\mu(x) - \hat\mu(x-1) } \quad \text{ for } \quad P_x = \frac{N_x}{\sum_{x \in 9 \ldots 12}N_x} \\ &= \sum_x \hat \alpha(x) \hat \mu(x) \qfor \hat\alpha(x) = \begin{cases} P_{12} & \text{ if } x = 12 \\ P_{x} - P_{x+1} & \text{ if } x \in \{9 \ldots 11\} \\ -P_9 & \text{ if } x = 8 \end{cases} \end{aligned} \]

Step 2. Estimator’s Expectation in Standard Form

\[ \begin{aligned} \mathop{\mathrm{E}}\qty[\sum_x \hat\alpha(x) \hat \mu(x)] &\overset{\texttip{\text{ \ ❓ \ }}{linearity of expectations}}{=} \sum_x \mathop{\mathrm{E}}\qty[\hat\alpha(x) \hat\mu(x)] \\ &\overset{\texttip{\text{ \ ❓ \ }}{law of iterated expectations}}{=} \sum_x \mathop{\mathrm{E}}\qty{ \mathop{\mathrm{E}}\qty[\hat\alpha(x) \hat \mu(x) \mid X_1 \ldots X_n] } \\ &\overset{\texttip{\text{ \ ❓ \ }}{linearity of conditional expectations}}{=} \sum_x \mathop{\mathrm{E}}\qty{ \hat \alpha(x) \mathop{\mathrm{E}}[\hat \mu(x) \mid X_1 \ldots X_n] } \\ &\overset{\texttip{\text{ \ ❓ \ }}{conditional unbiasedness of subsample means}}{=} \sum_x \mathop{\mathrm{E}}\qty{ \hat \alpha(x) \mu(x) } \\ &\overset{\texttip{\text{ \ ❓ \ }}{linearity of expectations}}{=} \sum_x \mathop{\mathrm{E}}\qty{ \hat \alpha(x) } \mu(x) \end{aligned} \]

  • This tells us we have an unbiased estimator if
  • the coefficients of our estimation target in standard form
  • are the expected value of our estimator’s coefficients.

Step 3. Dealing with Specifics

\[ \begin{aligned} &\text{Does} \quad \mathop{\mathrm{E}}\qty{ \hat \alpha(x) } = \alpha(x) \qfor &&\hat\alpha(x) = \begin{cases} P_{12} & \text{ if } x = 12 \\ P_{x} - P_{x+1} & \text{ if } x \in \{9 \ldots 11\} \\ -P_9 & \text{ if } x = 8 \end{cases} \\ \qqtext{and} &&&\alpha(x) = \begin{cases} p_{12} & \text{ if } x = 12 \\ p_{x} - p_{x+1} & \text{ if } x \in \{9 \ldots 11\} \\ -p_9 & \text{ if } x = 8 \end{cases} \end{aligned} \]

  • For this estimator, that question boils down to the unbiasedness of sample proportions. Almost.
  • These aren’t proportions of our whole sample. They’re proportions of the ‘high school subsample’.
  • But we we can show that \(\mathop{\mathrm{E}}[P_x]=p_x\) for all \(x\) via a sort of ‘two-stage sampling argument’.
    • First flip a coin to decide if \(X \in \{9 \ldots 12\}\) or not.
    • Then roll a die to decide which value you take within the group.
    • \(P_x\) is, conditional on the coin flip, a sample proportion.
    • So, it’s unbiased for the population proportion \(p_x\).

Beyond these two summaries

0.0 0.1 0.2 0.3 0.4 -2 0 2

  • We’ve shown that our two specific estimators are unbiased.
  • More generally, we’ve shown that a linear combination of subsample means is unbiased …
    • if our estimation target is a linear combination of subpopulation means …
    • … in which our estimator’s coefficients are replaced by their expected values.

\[ \begin{aligned} \mathop{\mathrm{E}}\qty[\sum_x \hat\alpha(x) \hat \mu(x) ] &= \sum_x \mathop{\mathrm{E}}[\hat\alpha(x)] \mu(x) \\ \end{aligned} \]

  • This can be used for all kinds of summaries.
    • Almost every target we’ll talk about this semester is a linear combination like this.
    • And the ones that aren’t, like ratios, can be approximated by linear combinations.

Variance

Review: The Law of Total Variance

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.00 0.25 0.50 0.75 1.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.00 0.25 0.50 0.75 1.00

\[ \mathop{\mathrm{\mathop{\mathrm{V}}}}[Y] = \mathop{\mathrm{E}}\qty{\mathop{\mathrm{\mathop{\mathrm{V}}}}( Y \mid X ) } + \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty{\mathop{\mathrm{E}}( Y \mid X ) } \]

  • The Law of Total Variance breaks Variance into within-group and between-group terms.
  • It’s like the Law of Iterated Expectations, but for Variance.
  • It’s a useful way to decompose the variance of a random variable.
  • Think about where most of the variance in \(Y\) is coming from in the two populations shown above.

A Variance Decomposition for our Estimator

\[ \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[\sum_x \hat\alpha(x) \hat\mu(x)] = \textcolor{blue}{\sum_x \sigma^2(x) \times \mathop{\mathrm{E}}\qty[ \frac{\hat\alpha(x)^2}{N_x} ]} + \textcolor{red}{\mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[ \sum_x \mu(x) \hat \alpha(x) ]} \qfor \sigma^2(x) = \mathop{\mathrm{\mathop{\mathrm{V}}}}[Y_i \mid X_i=x] \]

  • We can use the law of total variance to decompose our estimator’s variance into two parts.
    • The first reflects the randomness of our subsample means.
    • The second reflects the randomness of our coefficients \(\hat\alpha(x)\).
  • Note that if our coefficients aren’t random, i.e. \(\hat\alpha(x)=\mathop{\mathrm{E}}[\hat\alpha(x)]\), the second term is zero.
  • And often, even if they are random, the first term is much bigger.
  • As a result, we can usually get away with thinking about the first part only.
    • Note that the second part is positive—it’s the variance of something.
    • Ignoring it gives us a lower bound on the actual variance of our estimator.

The Formula

We won’t get deep into the derivation now. I’ll guide you through it in the homework.

\[ \small{ \begin{aligned} \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[ \sum_x \alpha(x) \hat \mu(x) ] &= \mathop{\mathrm{E}}\qty[ \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty{ \sum_x \hat\alpha(x) \hat \mu(x) \mid X_1 \ldots X_n }] \quad &&+ \quad \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[ \mathop{\mathrm{E}}\qty{\sum_x \hat\alpha(x) \hat \mu(x) \mid X_1 \ldots X_n} ] \\ &= \mathop{\mathrm{E}}\qty[ \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty{ \sum_x \hat\alpha(x) \hat \mu(x) \mid X_1 \ldots X_n }] \quad &&+ \quad \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[ \sum_x \hat\alpha(x) \mathop{\mathrm{E}}\qty{ \hat \mu(x) \mid X_1 \ldots X_n} ] \\ &= \mathop{\mathrm{E}}\qty[ \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty{ \sum_x \hat \alpha(x) \hat \mu(x) \mid X_1 \ldots X_n }] \quad &&+ \quad \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[ \sum_x \hat\alpha(x) \mu(x) ] \\ &= \textcolor{blue}{\sum_x \mathop{\mathrm{E}}\qty[ \hat\alpha(x)^2 \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty{\hat\mu(x) \mid X_1 \ldots X_n}]} \quad &&+ \quad \textcolor{blue}{\sum_x \sum_{x'} \mu(x) \mu(x') \mathop{\mathrm{Cov}}\qty[\hat\alpha(x),\ \hat\alpha(x')]} \\ &=\textcolor{blue}{ \sum_x \sigma^2(x) \times \mathop{\mathrm{E}}\qty[ \frac{\hat\alpha(x)^2}{N_x} ]} \quad &&+ \quad \textcolor{blue}{\sum_x \sum_{x'} \mu(x) \mu(x') \mathop{\mathrm{Cov}}\qty[\hat\alpha(x),\ \hat\alpha(x')]} \end{aligned} } \]

Variance Estimation

\[ \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[\sum_x \alpha(x) \hat \mu(x)] \ge \textcolor{blue}{\sum_x \sigma^2(x) \times \mathop{\mathrm{E}}\qty[ \frac{\hat\alpha(x)^2}{N_x} ] \quad \text{ where } \quad \sigma^2(x) = \mathop{\mathrm{\mathop{\mathrm{V}}}}[Y_i \mid X_i=x]} \]

  • LThe lower bound has two quantities we don’t know, but can estimate easily.

    • The subpopulation variances \(\sigma^2(x)\).
    • The expected value of the ratio of the coefficient squared and the sample size, \(\hat\alpha(x)^2 / N_x\).

  • The subsample variances are good esimates of the subpopulation variances \[ \sigma^2(x) \approx \hat\sigma^2(x) := \frac{1}{N_x} \sum_{i:X_i=x} \qty{Y_i - \hat\mu(x)}^2 \]

  • The squared coefficient / subsample size ratio is usually a good estimate of its expectation.

\[ \frac{\hat\alpha(x)^2}{N_x} \approx \mathop{\mathrm{E}}\qty[\frac{\hat\alpha(x)^2}{N_x}] \]

  • Plugging them in gives us an estimate of our summary’s variance lower bound.

\[ \mathop{\mathrm{\mathop{\mathrm{V}}}}[\hat\theta] \ge \sum_x \ \frac{\hat\sigma^2(x)\hat\alpha(x)^2}{N_x} \]

The Average Over Years

$0k $25k $50k $75k $100k $125k $150k $175k $200k 8y 9y 10y 11y 12y 13y 14y 15y 16y 17y 18y 19y 20y

\[ \hat\sigma_{\hat\theta}^2 = \sum_x \frac{\hat\alpha^2(x) \ \hat\sigma^2(x)}{N_x} \]

where

\[ \begin{array}{l|lll} x & 8 & 9 & 10 & 11 & 12 & 13 & \ldots & 20 \\ \hline \hat\alpha^2(x) & 0.06 & 0.00 & 0.00 & 0.00 & 0.06 & 0.00 & \ldots & 0.00 \\ \hat\sigma^2(x) & 492M & 444M & 113M & 603M & 803M & 1B & \ldots & 14B \\ N_x & 14 & 26 & 16 & 75 & 605 & 357 & \ldots & 71 \\ \frac{\alpha^2(x) \ \hat\sigma^2(x)}{N_x} & 2.20M & 0.00 & 0.00 & 0.00 & 83.01K & 0.00 & \ldots & 0.00 \\ \end{array} \]

Exercise. Use a caricature of this table to approximate our estimator’s variance.

Comparing Our Two Summaries

0e+00 1e-04 2e-04 -5000 0 5000 10000 15000
Interval Estimate for the Average over Years
0.00000 0.00005 0.00010 0.00015 -5000 0 5000 10000 15000
Interval Estimate for the Average over People

\[ \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[\sum_x \hat\alpha(x) \hat\mu(x)] \ge \mathop{\mathrm{E}}\qty[ \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty{ \sum_x \hat\alpha(x) \hat \mu(x) \mid X_1 \ldots X_n }] = \sum_x \sigma^2(x) \times \mathop{\mathrm{E}}\qty[ \frac{\hat\alpha(x)^2}{N_x} ] \]

  • Let’s think about what this means for our two examples. \[ \small{ \begin{aligned} \hat\theta_{\text{years}} &= \frac14\sum_{x=9}^{12} \qty{ \hat\mu(x) - \hat\mu(x-1)} = \sum_x \alpha(x) \hat \mu(x) \qfor \alpha(x) = \begin{cases} \hphantom{-}\frac14 & \text{ if } x = 12 \\ -\frac14 & \text{ if } x= 8 \\ 0 & \text{ otherwise} \end{cases} \\ \hat\theta_{\text{people}} &= \sum_{x=9}^{12} P_x \qty{ \hat\mu(x) - \hat\mu(x-1)} = \sum_x \hat \alpha(x) \hat \mu(x) \qfor \hat\alpha(x) = \begin{cases} \hphantom{-}P_{12} & \text{ if } x = 12 \\ P_{x} - P_{x+1} & \text{ if } x \in \{9 \ldots 11\} \\ -P_9 & \text{ if } x = 8 \end{cases} \end{aligned} } \]

  • We’ve seen that the variance of our estimate of the average-over-years summary has large variance.

  • The problem was that it paired a relatively large weight, \(\alpha(x)=-1/4\), with a very small subsample size \(N_8=14\).

  • And we know the average-over-people summary has an even larger variance. Let’s compare.

A Comparison

\[ \begin{array}{l|c|ccccc|c} & x & 8 & 9 & 10 & 11 & 12 & \sum \\ & N_x & 14 & 26 & 16 & 75 & 605 & 736 \\ & \hat\sigma^2(x) & 492M & 444M & 113M & 603M & 803M & \\ &\hline \\ \text{for}\ \ \hat\theta_{\text{people}} &\hat\alpha^2(x) & 0.00 & 0.00 & 0.01 & 0.52 & 0.68 & \\ &\frac{\hat\sigma^2(x)\alpha^2(x)}{N_x} & 44K & 3K & 45K & \textcolor{blue}{4.2M} & 897K & 5.2M \\ \hline \\ \text{for}\ \ \hat\theta_{\text{years}} &\hat\alpha^2(x) & 0.06 & 0.00 & 0.00 & 0.00 & 0.06 & \\ &\frac{\hat\sigma^2(x) \alpha^2(x)}{N_x} & \textcolor{blue}{2M} & 0 & 0 & 0 & 83K & 2M \\ \\ \end{array} \]

  • For both summaries, most of the variance comes from one term.
  • Let’s try to compare their variances ignoring all but the biggest term in each.
    • For the average over years, it’s the \(x=8\) subsample, with a sample size of \(N_x=14\) and a coefficient of \(-1/4\).
    • For the average over people, it’s the \(x=11\) subsample, with …
      • a sample size \(N_x=75\) that’s roughly 4x bigger.
      • but a coefficient of \(-0.72\) that’s roughly 3x bigger, too.
  • Variance grows with the square of the coefficient divided by the sample size.
  • So this approximation suggests variance would be roughly \(3^2 / 4 \approx 2.25x\) bigger for the average over people.
    • That’s not far off what we get when we compare bootstrap variance estimates.
    • That’s 2.2x bigger.
  • Intuitive Conclusion.
    • Summaries that involve the means of small groups are hard to estimate.
    • But that’s not all that matters. It matters how much they’re used.

What About The Other Part?

\[ \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[ \sum_x \alpha(x) \hat \mu(x) ] = \mathop{\mathrm{E}}\qty[ \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty{ \sum_x \hat\alpha(x) \hat \mu(x) \mid X_1 \ldots X_n }] \quad + \quad \mathop{\mathrm{\mathop{\mathrm{V}}}}\qty[ \mathop{\mathrm{E}}\qty{\sum_x \hat\alpha(x) \hat \mu(x) \mid X_1 \ldots X_n} ] \]

  • What we’ve been thinking about so far is the first part of the variance.
    • That’s zero when the coefficients are constants, but not when they’re random.
    • Because this part was larger for the summary with random coefficients, that implied its variance would be larger.
    • If it had been smaller, that wouldn’t have done much for us.
  • But to gain intuition about the term we left out, let’s approximate it.
    • We can use the bootstrap to estimate the variance of our summary—the whole thing.
    • We’ll use the variance (draws from) the bootstrap sampling distribution. It’s 5.24M.
    • Our estimate for the first part only — based on our table — was 5.16M.
    • That suggests that the second part should be roughly \(5.24M - 5.16M \approx 100K\).

Summary

  • When we’re estimating is a linear combination of the subpopulation means, the sample version is unbiased …
  • if the sample coefficients are unbiased estimates of the population ones.

\[ \small{ \hat \theta = \sum_x \alpha(x) \hat\mu(x) \quad \text{ satisfies } \mathop{\mathrm{E}}[\hat\theta] = \theta } \]

  • That’s something you have to think about case-by-case.

  • But proportions—even proportions of subsamples—are, in fact, unbiased.

  • Its variance is determined by the variance in each subpopulation, the coefficients, and the subsample sizes.

  • And we can estimate it. Or, at least, a lower bound. \[ \small{ \mathop{\mathrm{\mathop{\mathrm{V}}}}[ \hat\theta ] \ge \sum_x \sigma^2(x) \alpha^2(x) \times \mathop{\mathrm{E}}\qty[ \frac{1}{N_x} ] \approx \sum_x \hat\sigma^2(x)\alpha^2(x) \times \frac{1}{N_x} =: \hat\sigma_{\hat\theta}^2 } \]

  • Fundamentally, we’re going to get an imprecise estimate if a small subsample has a large weight.

  • By imprecise, we mean a large standard error and a wide confidence interval.

Formula vs. Bootstrap

\[ \small{ \mathop{\mathrm{\mathop{\mathrm{V}}}}[ \hat\theta ] \ge \sum_x \sigma^2(x) \times \mathop{\mathrm{E}}\qty[\frac{\hat\alpha^2(x)}{N_x} ] \approx \sum_x \hat\sigma^2(x) \times \frac{\alpha^2(x)}{N_x} =: \hat\sigma_{\hat\theta}^2 } \]

  • We can use this formula construct a confidence interval based on normal approximation.

\[ \small{ \theta \in \qty[ \hat \theta - 1.96\hat\sigma_{\hat\theta}, \ \hat \theta + 1.96\hat\sigma_{\hat\theta}] \quad \text{ with probability } \approx .95 \qfor \hat\sigma_{\hat\theta}^2 = \sum_x \hat\sigma^2(x)\times \frac{\hat\alpha^2(x)}{N_x} } \]

  • Or we could use the bootstrap. Which do you prefer? And why?