96 I.J.Myung Journal of Mathematical Psychology 47(2003)90-100 Power Expanential Data 0. > 0.5 0.4 0.3 0.2 0.1 2 B10 14 18 18 20 Time in Seconds Fig.4.Modeling forgetting data.Squares represent the data in Murdock (1961).The thick (respectively,thin)curves are best fits by the power (respectively,exponential)models. Table 1 Summary fits of Murdock(1961)data for the power and exponential models under the maximum likelihood estimation(MLE)method and the least- squares estimation (LSE)method. MLE LSE Power Exponential Power Exponential Loglik/SSE() -313.37(0.886) -305.31(0.963) 0.0540(0.894) 0.0169(0.967) Parameter wi 0.953 1.070 1.003 1.092 Parameter w2 0.498 0.131 0.511 0.141 Note:For each model fitted,the first row shows the maximized log-likelihood value for MLE and the minimized sum of squares error value for LSE. Each number in the parenthesis is the proportion of variance accounted for(i.e.r2)in that case.The second and third rows show MLE and LSE parameter estimates for each of w and w2.The above results were obtained using Matlab code described in the appendix. This quantity is to be maximized with respect to the yield the observed data (v1,....)=(0.94,0.77,0.40, two parameters,w and w2.It is worth noting that the 0.26,0.24,0.16),from which the number of correct last three terms of the final expression in the above responses,xi,is obtained as 100yi,i=1,...,6.In equation (i.e.,In n!-In(n-xi)!-Inxi!)do not depend Fig.4,the proportion recall data are shown as squares. upon the parameter vector,thereby do not affecting the The curves in Fig.4 are best fits obtained under MLE. MLE results.Accordingly,these terms can be ignored, Table 1 summarizes the MLE results,including fit and their values are often omitted in the calculation of measures and parameter estimates,and also include the the log-likelihood.Similarly,for the exponential model, LSE results,for comparison.Matlab code used for the its log-likelihood function can be obtained from Eq.(15) calculations is included in the appendix. by substituting wi exp(-w2ti)for wit2. The results in Table 1 indicate that under either In illustrating MLE,I used a data set from Murdock method of estimation,the exponential model fit better (1961).In this experiment subjects were presented with a than the power model.That is,for the former,the log- set of words or letters and were asked to recall the items likelihood was larger and the SSE smaller than for the after six different retention intervals,(t1,...,16)= latter.The same conclusion can be drawn even in terms (1,3,6,9,12,18)in seconds and thus,m=6.The of r2.Also note the appreciable discrepancies in proportion recall at each retention interval was calcu- parameter estimate between MLE and LSE.These lated based on 100 independent trials (i.e.n=100)to differences are not unexpected and are due to the factThis quantity is to be maximized with respect to the two parameters, w1 and w2: It is worth noting that the last three terms of the final expression in the above equation (i.e., ln n!  lnðn  xiÞ!  ln xi!) do not depend upon the parameter vector, thereby do not affecting the MLE results. Accordingly, these terms can be ignored, and their values are often omitted in the calculation of the log-likelihood. Similarly, for the exponential model, its log-likelihood function can be obtained from Eq. (15) by substituting w1 expðw2tiÞ for w1t w2 i : In illustrating MLE, I used a data set from Murdock (1961). In this experiment subjects were presented with a set of words or letters and were asked to recall the items after six different retention intervals, ðt1;y; t6Þ ¼ ð1; 3; 6; 9; 12; 18Þ in seconds and thus, m ¼ 6: The proportion recall at each retention interval was calcu￾lated based on 100 independent trials (i.e. n ¼ 100) to yield the observed data ðy1;y; y6Þ¼ð0:94; 0:77; 0:40; 0:26; 0:24; 0:16Þ; from which the number of correct responses, xi; is obtained as 100yi; i ¼ 1;y; 6: In Fig. 4, the proportion recall data are shown as squares. The curves in Fig. 4 are best fits obtained under MLE. Table 1 summarizes the MLE results, including fit measures and parameter estimates, and also include the LSE results, for comparison. Matlab code used for the calculations is included in the appendix. The results in Table 1 indicate that under either method of estimation, the exponential model fit better than the power model. That is, for the former, the log￾likelihood was larger and the SSE smaller than for the latter. The same conclusion can be drawn even in terms of r2: Also note the appreciable discrepancies in parameter estimate between MLE and LSE. These differences are not unexpected and are due to the fact Fig. 4. Modeling forgetting data. Squares represent the data in Murdock(1961). The thick(respectively, thin) curves are best fits by the power (respectively, exponential) models. Table 1 Summary fits of Murdock(1961) data for the power and exponential models under the maximum likelihood estimation (MLE) method and the least￾squares estimation (LSE) method. MLE LSE Power Exponential Power Exponential Loglik/SSE ðr2Þ 313:37 ð0:886Þ 305:31 ð0:963Þ 0.0540 (0.894) 0.0169 (0.967) Parameter w1 0.953 1.070 1.003 1.092 Parameter w2 0.498 0.131 0.511 0.141 Note: For each model fitted, the first row shows the maximized log-likelihood value for MLE and the minimized sum of squares error value for LSE. Each number in the parenthesis is the proportion of variance accounted for (i.e. r2) in that case. The second and third rows show MLE and LSE parameter estimates for each of w1 and w2: The above results were obtained using Matlab code described in the appendix. 96 I.J. Myung / Journal of Mathematical Psychology 47 (2003) 90–100