5.3 Estimation of ratios of population means

Suppose we know the population mean $\mu_x$ of an auxiliary variable $x$, and it is associated with the variable of interest, $y$. To estimate the population means of the target variable $\mu_y$, intuitively, we could first estimate the ratio of their population means, $R_{y,x} $ and multiply it to $\mu_x$, such that $\hat\mu_{yR}= \hat R_{y,x} \mu_x$, where the subscript “$R$” denotes “Ratios”, see (Wu and Thompson 2020). In this chapter, we demonstrate how to estimate the ratio between the population means of two variables. We created a variable $\texttt{HWT_DHT_M_TRM_sq}$, which is the square of the self-reported height (in meters). The estimate of the ratio between the population mean of $\texttt{HWT_WGHT_KG_TRM}$ and that of $\texttt{HWT_DHT_M_TRM_sq}$ is computed as

\[ \frac{\sum_{i\in \mathcal{S}} w_i (HWT\_WGHT\_KG\_TRM_i) }{\sum_{i\in \mathcal{S}} w_i (HWT\_DHT\_M\_TRM\_sq_i )}, \] where $w_i$ is the survey weight and $\mathcal{S}$ is the set of sampled units.

svyratio(numerator = ~HWT_WGHT_KG_TRM, denominator = ~HWT_DHT_M_TRM_sq, 
         design = CLSA.design )

SAS

PROC SURVEYMEANS data = CLSAData   ratio;   
VAR HWT_WGHT_KG_TRM HWT_DHT_M_TRM_sq;                    
RATIO  HWT_WGHT_KG_TRM / HWT_DHT_M_TRM_sq ;
STRATA GEOSTRAT_TRM;                       
WEIGHT WGHTS_INFLATION_TRM;                                   
RUN;

SPSS

Click “Analyze” $\rightarrow$ “Complex Samples” $\rightarrow$ “Ratios…” $\rightarrow$ Browse and select the survey design file, “$\texttt{CLSADesign.csaplan}$” $\rightarrow$ Click “$\texttt{Continue}$” and enter variables “$\texttt{HWT_WGHT_KG_TRM}$” and “$\texttt{HWT_DHT_M_TRM_sq}$” to the the “Numerators” and “Denominator” panels, respectively $\rightarrow$ Click “Statistics…” $\rightarrow$ Select “Standard error” $\rightarrow$ Click “Continue” $\rightarrow$ Click “OK”.

Stata

svy linearized : ratio (HWT_WGHT_KG_TRM / HWT_DHT_M_TRM_sq)

Result comparison

Estimate	R	SAS	SPSS	Stata
Estimate	27.6076	27.6076	27.6076	27.6076
SE	0.3177	0.3177	0.3177	0.3177

Reference

Wu, Changbao, and Mary E. Thompson. 2020. Sampling Theory and Practice. Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-030-44246-0.