18.2 Effects of Nonresponse

Textbook example: Example reported in W.G. Cochran, Sampling Techniques, 3rd edition, 1977, ch. 13

Consider data that come form an experimental sampling of fruit orcharts in North Carolina in 1946. Three successive mailings of the same questionnaire were sent to growers. For one of the questions the number of fruit trees, complete data were available for the population…

Ave. # trees	# of growers	% of pop’n	Ave # trees/grower
1st mailing responders	300	10	456
2nd mailing responders	543	17	382
3rd mailing responders	434	14	340
Nonresponders	1839	59	290
	——–	——–	——–
Total population	3116	100	329

• The overall response rate was very low.
• The rate of non response is clearly related to the average number of trees per grower.
• The estimate of the average trees per grower can be calculated as a weighted average from responders \(\bar{Y_{1}}\) and non responders \(\bar{Y_{2}}\).

Bias: The difference between the observed estimate \(\bar{y}_{1}\) and the true parameter \(\mu\).

\[ \begin{aligned} E(\bar{y}_{1}) - \mu & = \bar{Y_{1}} - \bar{Y} \\ & = \bar{Y}_{1} - \left[(1-w)\bar{Y}_{1} - w\bar{Y}_{2}\right] \\ & = w(\bar{Y}_{1} - \bar{Y}_{2}) \end{aligned} \]

Where \(w\) is the proportion of non-response.

• The amount of bias is the product of the proportion of non-response and the difference in the means between the responders and the non-responders.
• The sample provides no information about \(\bar{Y_{2}}\), the size of the bias is generally unknown without information gained from external data.