GERRODETTE ET AL.: CONFIDENCE LIMITS FOR POPULATION PROJECTIONS 



Table 5.— Results of the Monte Carlo simulation of the 6 time- 

 step projection of the population whose age structure and vital 

 rates are given in Table 1 (program SLT). Sample size for the 

 simulation was 5,000 trials. Results in this table should be com- 

 pared with the "predicted" values in the last row of Table 3. 

 Here P is the proportion of final population sizes greater than 

 the initial population size. 



as anticipated. Both the logarithmic transformation 

 (Fig. IB) and the root transformation (Fig. IC) 

 appear to normalize the distribution. When the 

 cumulative frequency distributions are plotted on 

 normal probability scales (dots in Fig. 1), however, 

 the root transformation appears superior to the 

 logarithmic. The dots in Figure IC are nearly linear, 

 indicating that the distribution is close to normal. 



In Table 6 the accuracy of the 95% confidence 

 limits for the total population size computed by the 

 logarithmic and root transformations is compared 

 for projections of 2, 5, and 10 time steps, using the 

 same numerical example. Program SLT calculates 

 the proportion of final populations which fall above 

 and below the computed upper and lower confidence 

 limits. We expect that 2.5% of the cases should fall 

 above the upper limit and 2.5% below the lower limit 

 if the 95% confidence interval has been correctly 

 estimated. Table 6 shows that both the logarithmic 

 and the root transformations do a fair job of esti- 

 mating the 95% confidence limits. The root transfor- 

 mation, however, appears more accurate in this 

 example, as well as in other examples we have tried, 

 when the number of time steps is small. When the 

 number of time steps is large (50-100), both transfor- 

 mations produce normally distributed variables. 



Since the root transformation gave the most accu- 

 rate results for short projections, we used this trans- 

 formation in program SPP to compute a confidence 

 interval on total population size. More details of the 



Figure 1 .- Distributions of future total population size under 

 variable conditions. Histograms show the percentage frequency, 

 and dots the cumulative percentage frequency plotted on a normal 

 probability scale, for 5,000 stochastic projections of the population 

 given in Table 1 for six time steps. A. Distribution of NqINq, the 

 final population size divided by the initial. B. Distribution of logg 



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