False

If N_d were known, then we would want to use SYS instead.

Numerator variable u = data value for domain, and 0 if otherwise (slide 12 week 11) y_i if i in U_d, 0 if otherwise

Denominator variable x = indicator of domain membership.  1 if i in U_d, 0 if otherwise

H₀: ybar_U1 = ybar_U2

H₁: ybar_U1 = ybar_U2

Equivalently:

H₀: ybar_U1 - ybar_U2 = 0

H₁: ybar_U1 - ybar_U2 = 0

z = (ybar₁ - ybar₂)/sqrt(V(ybar₁)+V(ybar₂))

Reject H₀ if abs(z) > z_alpha/2

Potential bias

Loss of precision

Design phase.

After data collection: call-backs, post-stratification, impute

Call-backs fix both unit and item non-response.

Post-stratification - unit non-response primarily.

Imputation - item non-response primarily.

Unit 

Item

n_R/n

 where n_R = realized sample size

durb

Differences between population means

Nonresponse rate

Sample size reductions due to NR affect precision by increasing variances.

Remedy by anticipating and designing for NR sample size attrition

Method:  divide the target sample size desired (n) by the guessed proportion of respondents (N_R/N: R).  Formula: n/R

Select a sample from the nonrespondents to the survey.

Collect data from contacted nonrespondents.

Use these data to estimate population mean for nonrespondents ybar_MU.

Estimate populaiton mean for whole population ybar_U with a weighted combination of respondent sample mean and nonrespondent sample mean. 

Divide population into H mutually exclusive and exhaustive post-strata.

For each post-stratum: Know post-stratum sizes N_h, estimate characteristics of post-strata, use post-stratum sample mean to estimate post-stratum population mean, pool post-stratum estimates using, for example, a weighted mean of the post-stratum estimates.

A statistical method for "filling in" or "predicting" missing values.

Impute values so that they represent the distribution of the response variable with missing data (y).

Impute values using a method that supports estimation of the variance associated with the random components of the imputation process.

Deductive imputation

Call mean imputation

hot-deck imputation (random)

regression imputation

multiple imputation

common method, rarely implementable

Use a deterministic rule to assign a value (e.g. crime victim: no = violent crime victim: no)

There must be sufficien nformation to identify the missing value with a high degree of certainty.

Relatively uncommon, especially with use of computer-assisted survey instruments when checks for these realtionships are embedded inteh computer-based questionnaire.

Avoid: leads to incorrect distribution of y in dataset.

Divide responding units in to imputation classes.

With a given imputation class: calculate the average value for available item data in class, fill in missing value for nonresponding unit with average value.

Retains mean estimate for an imputation class. Underestimates variance within an imputation class, which misrepresents distribution of y.

Most common and generally applicable.

May apply within groups of respondents (auxilliary info).

Divide responding units in to imputation classes.  Within a given imputation class: randomly select a donor from responding units in class, filling in missing value for nonresponding unit with value from donor unit.

Retains variation  in individual values, can impute from many variables from same donor, variations exist

Uses model to incorporate auxiliary information, between hot-deck and cell mean imputation methods.

Use a regression model to relate covariate(s) to variable with missing data.

Estimate regression parameters with data from responding units, fill in missing value with predicted value.

Useful if a strong relationship exists that provides a better predicted value for the missing data, form of (conditional) mean imputation, requires separate model for each variable with missing data.

Accounting for variation due to imputation process.

Decide on an imputation model, impute m>1 values for each missing data item, result is m (different) data sets with no missing values.

Variation in estimates across data sets provides an estimate of the variability associated with the imputation process, analysis is more complex.

A cluster sample is a probability sample in which a sampling unit is a cluster.

We will no longer assume SU = element 

Divide the population (of K elements) into N total clusters.

Take a sample of n clusters.

1-stage CS:  A block of cells is a cluster, SU is a cluster, don't sample from every cluster.

STS: A block of cells is a stratum, SU is an element, sample from every stratum.

May not have a list of elements for a frame, but a list of clusters may be available.

May be cheaper to conduct the study if elements are clustered.

Elements within clusters tend to be correlated due to exposure to similar conditions.

We get less information than if we observe the same number of unrelated elements.

Define clusters for which within-cluster variation is high (rarely possible).

Define clusters that are relatively small.

i = index for cluster i

i, j = index for element j in cluster i

N = total clusters in population

n = sampled clusters

M_i = elements in a cluster

K = number of elements in population (sum M_i)

N/n

Yes, it is self-weighting.

(N/n)*(M_i/m_i)

It is not always self-weighting.

ybar_iU = t_iU/M_i

S_i² = 1/(M_i - 1)*sum[(y_ij - ybar_iU)²]

Q_i/nψ_i = Q_iK/nM_i

Not always self weighting (??)

k/nm_i

Not always self weighting depending on m_i (??)

Cluster id (i)

Element id within cluster (j)

Variable (y_ij)

Cluster id (i)

Cluster total under 1-stage CS (t_iU)

Cluster mean under 1-stage CS (ybar_iU)

Within-cluster variance under 1-stage CS (s_i²)

Usually t_i (cluster total) is positively correlated with M_i (cluster size)

No intercept

Notation of chapter 3 versus notation of chapter 5 ratio:  y_i (variable of interest) = t_i (cluster total), x_i (auxiliary info) = M_i (cluster size)

The average cluster size for population

If unknown, can estimate with sample mean of cluster sizes Mbar_s = 1/n*sum(M_i)

Stage 1: Select clusters.  SRSWOR of n PSUs from population of N PSUs.

Stage 2: Select elements within each sampled cluster. SRSWOR of m_i SSUs from M_i elements in PSU i sampled in stage 1.

First stage sampling unit is a primary sampling unit (PSU) = cluster.

Second stage sampling unit is a secondary sampling unit (SSU) = element

Only collect data on the SSUs that were sampled from the cluster.

Likely that elements in cluster will be correlated.

-May be inefficient to observe all elements in a sample PSU and the extra effort required to fully enumerate a PSU does not provide that much extra information.

May be better to spend resources to sample many PSUs and a small number of SSUs per PSU. (Possible opposing force: study costs associated to going to many clusters)

1. Variation among PSUs

2. Variation amonog SSUs within PSUs

[image]

False

It does NOT imply equal inclusion probability for an element (unconditional probability for element)

slide 65 of week 13

Unbiased estimation - Use if you know K or N

e.g. N= total number of clutches or K = total number of eggs in Minnedosa, Manitoba

Ratio estimation - Only requires knowledge of M_i (e.g. number of eggs in clutch i), in addition to data collected

When cluster sizes (M_i) are unequal

t_i (cluster total) is roughly proportional to M_i (cluster size)

When t_i is roughly proportional to M_i (bigger cluster = larger t_i)

This happens frequently in pops where cluster sizes (M_i) vary

π_i = P{cluster i in sample} = n/N

π_j|i = Pr{element j GIVEN cluster i in sample} = m_i/M_i

πij = Pr{element j AND cluster i in sample} =  π_iπ_j|i = (n/N)x(m_i/M_i) = nm_i/NM_i

Stage 1: Select n PSUs from N PSUs in pop using SRS

Stage 2: Choose m_i proportional to M_i so that m_i/M_i is constant, use SRS to select sample - if this is achieved, then it is self-weighting

Sample weight for SSU j in cluster i is constant for all elements.  Weight may vary slightly in practice, however, because it may not be possible for m_i/M_i to be equal to 1/c for all clusters.

They are appealing because you can use the simple mean estimator.

The caveat for variance estimation is that there is not break on variance of estimator - must use proper variance estimation formula for sample design.

SRS

SYS

STS with proportional allocation

CSE1

CSE2 with m_i proportional to M_i or c = M_i/m_i

Select cluster with PPSWR (stage 1)

Select elements with SRSWOR (stage 2)

Unbiased

Variance estimator is also unbiased.

Also holds for population mean.

True

Because we use WR sampling, variance estimator captures both between and within cluster variance.

This holds for the estimator for the population mean.