Estimands, Estimators, and Estimates

There’s the Average/Individual/Population Causal Effect as these vaunted things we want. These are estimands. You are going to use estimators to try and estimate them. Yeah quite the mouthful.

Term	Meaning	Example
Estimand	The true quantity we want to know1	Average causal effect among 1,000 patients
Estimator	The mathematical procedure we will use to estimate it	Difference in observed group means
Estimate	The numerical result after applying the estimator	0.15, or 15 percentage points

Unbiasedness in Estimators

Super helpful for peace of mind 💆‍♀️ You do the math and try and prove that ‘something’ is an unbiased estimator of the truth, the estimand.

Famous one: The sample mean is an unbiased estimator of the true population mean, the estimand². So now you can have a more concrete table:

Concept	Meaning	Mathematical notation
Estimand	The true population mean we want to know	$\mu$
Estimator	The sample mean we use to estimate it	$\bar{Y}$
Estimate	The numerical value calculated from our particular sample	e.g., 170 cm

Lovely stuff.

Going Causal

Let’s look at the SACE:

$$\begin{align*} SACE &= \frac{1}{N}\sum_{i=1}^{N}(Y_i^1 - Y_i^0) && \text{(Estimand)} \\ \widehat{SACE} &= \bar{Y}_{1,\mathrm{obs}} - \bar{Y}_{0,\mathrm{obs}} && \text{(Estimator)} \end{align*}$$

In a completely randomized experiment,

$$E_A[\widehat{SACE}] = SACE$$

In English: we are averaging over all possible randomized treatment assignments, holding the study participants fixed (read that again! ‘Sample’ in SACE!)

Identification and Estimation

Yeah we can do a table…

	Identification	Estimation
Question	Can the causal effect be recovered?	What is its numerical value?
Depends on	Assumptions and study design	Data and statistical procedures
In God Mode, with unlimited data?	Still might be impossible	Removes sampling uncertainty under suitable conditions
E.g.	Showing $E[Y^1] = E[Y \mid A=1]$	Calculating the treated group’s sample mean

…but that’s not the point. It is this flow:

$$\begin{gathered} \textbf{1. Define the estimand}\\ ACE = E[Y^1] - E[Y^0]\\ \downarrow\\ \textbf{2. Establish identification}\\ \text{Use assumptions to express the ACE in terms of observed data.}\\ \downarrow\\ \textbf{3. Estimate the effect}\\ \widehat{ACE} = \overline{Y}_{A=1} - \overline{Y}_{A=0} \end{gathered}$$

The first two steps are theoretical. It is only in the last step that we’re actually calculating something.

Footnotes

1. Remember God Mode… ↩

2. I.e. $E[\hat{Y}] = \mu$. If we repeatedly took random samples from the population and calculated their means, the average of those sample means would equal the true population mean. A single one won’t. Stop it. ↩