Appendix 4 - Assessment of statistical error in woodfuel consumption estimates

Table A1.1 - Statistical Results of Woodfuel Survey by Three Factors - Village, Wealth and Household Size

Effects	Sum of Squares	Degrees of Freedom	Mean Square	F-Statistic
SUMMER
Village	50.34	4	12.58	27.01+
Wealth	1.54	2	0.77	1.65*
Household size	125.28	2	62.64	134.42+
Error	100.67	216	0.466
WINTER
Village	40.33	4	10.08	6.72+
Wealth	4.21	2	4.61	3.07**
Household size	193.38	2	96.69	64.46+
Error	376.97	252	1.50

* Statistically significant at the 25% level
** Statistically significant at the 5% level
+ Statistically significant at the 0.1% level.

For each village/household size/wealth category, in winter and summer, a mean and sample variance were calculated for per capital woodfuel consumption. The variances of the sampling distributions of these means were estimated by division of the calculated sample variances by the number of observations in the appropriate category. In nearly all categories, these estimates were small. The major source of sampling error came from the differences in the sample means, for different villages, of the same household size/wealth categories. In the forecasting model, average per capital woodfuel consumption in each household size/wealth category was estimated by taking an arithmetical average of the means, for each category, for the five villages. The variance of the sampling distribution of each such mean was estimated by dividing the calculated sample variance by the number of villages, which was usually five. This procedure assumes that the village/household-size/wealth category sample means are random observations.

This procedure was used for the summer and winter data. If two independent random samples are taken from populations with distributions (1, 1²) and (2, 2²), for the combination of sample means ₁₊₂, the sampling distribution is (1 + 2, ² 1² + ² 2²). Thus, the winter and summer sample means were combined using weights that represented the lengths of the two seasons. The relative weights were one- and two-thirds respectively. Thus, the relative weights for the combination of the estimated variances of the sampling distributions were one- and four-ninths respectively.

This appendix examines the forecasting model for woodfuel consumption in the light of the survey data. The construction of the model is justified and a representative confidence interval is calculated for forecasts.

The basis of the survey and random selection of the five villages was described in Section 2.2.

The 50-60 households sampled in each village were placed in wealth classes - rich, medium and poor - and household size classes - small (1-4), middle (5-8) and large (9+). The means of household per capita woodfuel consumption in each household size/wealth category was calculated for each village and subsequently used as input to the forecasting model.

The usefulness of this classification of categories was tested by a three-factor analysis of variance investigation. The three factors were village, household size and wealth. As Scheffé has discussed, multi-factor analysis of variance is particularly complicated in the case of unequal numbers of observations in the different categories. In order to obtain more efficient estimates of the means within the categories with larger proportions of the population, the survey was deliberately designed to have more observations in these categories, representing large classes within the village. The resulting complex analysis of variance problem was resolved by following a simplifying, and approximately valid, suggestion of Scheffé.

If the underlying parent populations are normal, category sample-means and sample-variances are independent. Thus, the hypothesis of no factors having any effects can be tested against the hypothesis of one particular factor having an effect. Using standard notation, and denoting the particular factor by 'i' and others by 'j', 'k', and 'l' the effects equation is:

It is then possible to test the hypothesis

H_o: a _i = o (whatever value of i) and constant (whatever value of j, k, l).

where = variance of population of ijk category.

The statistical results are given in the table overleaf. The results strongly support the hypothesis that village, household size and wealth factors are determinants of household per capita woodfuel consumption. Thus, this part of the construction of the forecasting model is supported by the evidence of the survey.

There are potentially many sources of error in these forecasts made in Section 4.1. The assumptions underlying the economic and demographic projections and the structure of the model are probable significant sources of error. The errors from conventional sampling errors are relatively easy to quantify and calculate.

The estimated 1981 distribution of the proportions of the population in different household size/wealth categories was considered to be representative of the different regions' distributions. Using these proportions as weights for the estimates of per capita woodfuel consumption in the different categories, an estimate of the variance of the sampling distribution of the overall mean, per capita consumption of woodfuel was calculated. Assuming a normal sampling distribution, this gives a 95% confidence interval of ±12%.