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 $\text{HWT\_DHT\_M\_TRM\_sq}$, which is the square of the self-reported height (in meters). The estimate of the ratio between the population mean of $\text{HWT\_WGHT\_KG\_TRM}$ and that of $\text{HWT\_DHT\_M\_TRM\_sq}$ is computed as

$$\frac{{\sum }_{i\in S}{w}_i\left(HWT\_WGHT\_KG\_TR{M}_i\right)}{{\sum }_{i\in S}{w}_i\left(HWT\_DHT\_M\_TRM\_s{q}_i\right)},$$ where $w_i$ is the survey weight and $S$ is the set of sampled units.

R

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” $\to$ “Complex Samples” $\to$ “Ratios…” $\to$ Browse and select the survey design file, “$\text{CLSADesign.csaplan}$” $\to$ Click “$\text{Continue}$” and enter variables “$\text{HWT\_WGHT\_KG\_TRM}$” and “$\text{HWT\_DHT\_M\_TRM\_sq}$” to the the “Numerators” and “Denominator” panels, respectively $\to$ Click “Statistics…” $\to$ Select “Standard error” $\to$ Click “Continue” $\to$ 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.