Ratio and Regression Method of Estimation

 

2.4 Ratio Method

 Concept and rationale of Auxiliary Variable

An auxiliary variable is an additional variable that is related to the main variable of interest in a statistical study but is not directly of interest itself. It is often used to improve the efficiency of an estimation process. The auxiliary variable is known or observed alongside the variable of interest and helps in reducing the variance of the estimates when used appropriately.

The ratio method is most effective in situations where the auxiliary variable is easy to measure, and there is a strong linear relationship between the auxiliary and study variables. The method works best when the ratio of the two variables remains relatively stable across the population.

Rationale for Using Auxiliary Variables

Improving Precision: Auxiliary variables can significantly reduce the variance of the estimators, leading to more precise estimates. This is particularly useful when direct measurements of the main variable are difficult, costly, or prone to error.

Reducing Sampling Error: In the presence of a strong correlation between the auxiliary and the study variable, auxiliary information can reduce sampling error, thus making the estimates more reliable.

Bias Reduction: In some cases, the use of auxiliary information can help in adjusting for biases that arise due to non-response, under-coverage, or other survey issues.

Cost Efficiency: Since collecting data on the auxiliary variable may be easier or cheaper than on the variable of interest, it allows researchers to gather more information without significantly increasing costs.

Use in Estimation

Ratio Estimation: When the auxiliary variable is highly correlated with the study variable, ratio estimators can be used. The basic idea is to estimate the ratio of the variable of interest to the auxiliary variable and then use this ratio to estimate the population total or mean.

Example: Suppose you want to estimate the average income (study variable) of a population, and you know the population size (auxiliary variable). If there's a strong relationship between income and population size, you can use the ratio estimator to improve the precision of your income estimate.

Regression Estimation: Regression estimation involves using the auxiliary variable in a regression model to estimate the variable of interest. This approach assumes that the study variable can be predicted from the auxiliary variable based on a regression relationship.

Example: If crop yield (study variable) is being estimated using rainfall data (auxiliary variable), the regression estimation technique can leverage the linear relationship between the two to get a more accurate yield estimate.

Situations where the ratio method is appropriate

1. Agricultural Surveys

             While estimating the total crop yield in a region, total cultivated land area is auxiliary variable. There is often a stable, positive relationship between the amount of land cultivated and the crop yield. Using the ratio of yield to land area can provide an accurate estimate of total crop production.

2. Population Studies

             While estimating the total population of a town based on household data, number of households is auxiliary variable. If there is a stable average household size, the ratio of population to the number of households can be used to estimate the total population, especially if household counts are easier to obtain than individual population data.

3. Economic Surveys

While estimating the average income of a population based on employment data, number of employed individuals is auxiliary variable. There is often a direct relationship between the number of employed individuals and the total income in a population. The ratio method can use employment figures (which may be more accessible) to estimate income levels.

4. Industrial Production Estimation

While estimating the total output of a factory based on machine hours, machine operating hours is auxiliary variable. If there is a stable production rate per machine hour, the ratio of total output to machine hours can be used to estimate the overall production.

5. Environmental Studies

While estimating the total amount of water used in a region based on population size, the population size is auxiliary variable.   There is often a direct relationship between the population size and the amount of water used. Using the ratio of water use per person can help estimate the total water consumption.

6. Retail and Commerce

While estimating total sales for a company based on the number of transactions, number of transactions or customers is auxiliary variable.  There is usually a relationship between the number of customers or transactions and total sales. If the average sale per customer is stable, total sales can be efficiently estimated.

7. Education Surveys

While estimating the total number of students in a district based on the number of schools, number of schools or classrooms is auxiliary variable. There is often a stable student-to-school or student-to-classroom ratio, so this method can help estimate the total number of students in the district.

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2.5 Regression Method

Regression estimation is a statistical method used to estimate a population parameter (typically the mean or total) based on the relationship between the variable of interest (Y) and an auxiliary variable (X). This method is particularly useful when there is a strong linear relationship between the two variables.

The ratio method of estimation uses the auxiliary information which is correlated with the study variable to improve the precision.  When the regression of Yon X is linear, it is not necessary that the line should always pass through the origin. Under such conditions it is more appropriate to use the regression type estimators to estimate the population means.

 Situations where the regression method of estimation is appropriate:

    The regression method of estimation is appropriate when:

• There is a strong linear relationship between the study variable Y (the variable of interest) and an auxiliary variable X (which is known for all units in the population).
• The auxiliary variable X is correlated with Y and is easy or less costly to measure.
• The relationship between Y and X can be described by a regression line,

   i.e., Y=α +β X+ e  , where e represents random error.

This method is particularly useful in surveys where it is possible to collect X more reliably or inexpensively than Y, and the goal is to improve the precision of estimates for Y.

 

                                 

 Bias in Regression

The regression estimator is unbiased for the population mean and total if the regression model is correctly specified. However, if the true relationship between y and x is nonlinear, or if there are other omitted variables that affect y, the regression estimator may be biased. They are generally asymptotically unbiased, meaning that their bias decreases as the sample size increases. However, for small sample sizes or non-linear relationships, bias may occur.

 Relative Efficiency of Regression Estimators

The relative efficiency of an estimator measures how much better it performs compared to other estimators in terms of the variance (or mean square error).

     Comparison with SRSWOR (Simple Random Sampling without Replacement):

            The regression estimator is generally more efficient than the estimator obtained using SRSWOR if there is a strong correlation between the study variable YYY and the auxiliary variable XXX. The variance of the regression estimator is smaller than that of the SRSWOR estimator, especially when the auxiliary variable helps explain much of the variability in the study variable.

Comparison with the Ratio Estimator:

The regression estimator is often more efficient than the ratio estimator because it accounts for both the intercept and slope in the relationship between Y and X, whereas the ratio estimator only assumes a proportional relationship (no intercept). However, if the relationship is strictly proportional (i.e., passing through the origin), the ratio estimator might be as efficient as or even more efficient than the regression estimator.

In general, if the auxiliary variable X is linearly related to Y and the slope is not close to 1, the regression estimator is more efficient than both the SRSWOR and ratio estimators.

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