Sweating the Small Stuff: Does data cleaning and testing ... - Frontiers

Sheng and ShengEffect of non-normality on coefficient alphaTable 2 | Root mean square error and bias for estimating α for the simulated situations where the true score (t i ) distribution is normal ornon-normal.n k RMSE biasdist1 dist2 dist3 dist4 dist5 dist6 dist1 dist2 dist3 dist4 dist5 dist6ρ ′ XX = 0.330 5 0.251 0.244 0.308 0.252 0.247 0.305 −0.070 −0.066 −0.102 −0.070 −0.069 −0.10010 0.240 0.233 0.296 0.241 0.234 0.292 −0.069 −0.067 −0.101 −0.071 −0.067 −0.09830 0.232 0.226 0.287 0.232 0.226 0.288 −0.069 −0.067 −0.101 −0.070 −0.067 −0.10150 5 0.182 0.178 0.224 0.183 0.178 0.224 −0.048 −0.047 −0.067 −0.047 −0.046 −0.06810 0.173 0.167 0.216 0.173 0.169 0.215 −0.048 −0.044 −0.068 −0.048 −0.046 −0.06730 0.166 0.162 0.213 0.167 0.164 0.210 −0.046 −0.046 −0.069 −0.048 −0.046 −0.067100 5 0.122 0.120 0.154 0.123 0.121 0.154 −0.031 −0.031 −0.042 −0.032 −0.031 −0.04210 0.116 0.114 0.149 0.116 0.114 0.148 −0.032 −0.031 −0.043 −0.031 −0.031 −0.04230 0.112 0.109 0.147 0.113 0.110 0.147 −0.031 −0.030 −0.043 −0.032 −0.031 −0.0441000 5 0.040 0.040 0.052 0.041 0.040 0.051 −0.018 −0.018 −0.019 −0.018 −0.018 −0.01910 0.038 0.038 0.050 0.039 0.038 0.050 −0.018 −0.018 −0.019 −0.018 −0.018 −0.02030 0.038 0.037 0.049 0.038 0.037 0.049 −0.018 −0.018 −0.019 −0.018 −0.018 −0.019ρ ′ XX = 0.630 5 0.151 0.132 0.268 0.151 0.139 0.266 −0.051 −0.044 −0.121 −0.051 −0.046 −0.11810 0.142 0.125 0.261 0.145 0.131 0.261 −0.050 −0.043 −0.120 −0.051 −0.046 −0.12030 0.139 0.120 0.260 0.140 0.126 0.258 −0.050 −0.043 −0.120 −0.051 −0.045 −0.11950 5 0.109 0.097 0.198 0.110 0.101 0.198 −0.037 −0.033 −0.083 −0.037 −0.035 −0.08310 0.104 0.092 0.195 0.105 0.096 0.193 −0.037 −0.033 −0.084 −0.037 −0.034 −0.08330 0.100 0.088 0.193 0.102 0.092 0.191 −0.037 −0.033 −0.084 −0.037 −0.034 −0.083100 5 0.075 0.067 0.139 0.076 0.070 0.137 −0.028 −0.026 −0.055 −0.028 −0.027 −0.05410 0.071 0.063 0.137 0.072 0.066 0.135 −0.028 −0.026 −0.056 −0.028 −0.027 −0.05330 0.069 0.061 0.135 0.070 0.064 0.133 −0.028 −0.026 −0.055 −0.028 −0.027 −0.0541000 5 0.029 0.028 0.049 0.029 0.028 0.049 −0.020 −0.020 −0.024 −0.020 −0.020 −0.02410 0.028 0.027 0.048 0.029 0.027 0.048 −0.020 −0.020 −0.023 −0.020 −0.020 −0.02430 0.028 0.026 0.048 0.028 0.027 0.048 −0.020 −0.020 −0.024 −0.020 −0.020 −0.024ρ ′ XX = 0.830 5 0.078 0.060 0.197 0.080 0.066 0.198 −0.030 −0.022 −0.099 −0.030 −0.024 −0.09710 0.074 0.056 0.194 0.076 0.062 0.196 −0.029 −0.023 −0.098 −0.030 −0.024 −0.09930 0.072 0.053 0.193 0.074 0.060 0.193 −0.029 −0.022 −0.099 −0.029 −0.024 −0.09850 5 0.057 0.045 0.142 0.058 0.049 0.142 −0.022 −0.018 −0.067 −0.023 −0.020 −0.06710 0.054 0.042 0.140 0.055 0.046 0.140 −0.022 −0.018 −0.067 −0.022 −0.020 −0.06730 0.052 0.040 0.140 0.054 0.044 0.139 −0.022 −0.018 −0.068 −0.022 −0.019 −0.067100 5 0.039 0.032 0.096 0.040 0.035 0.095 −0.018 −0.016 −0.043 −0.018 −0.016 −0.04310 0.038 0.030 0.095 0.038 0.033 0.094 −0.018 −0.016 −0.043 −0.018 −0.016 −0.04330 0.037 0.029 0.094 0.037 0.032 0.094 −0.018 −0.016 −0.043 −0.018 −0.016 −0.0431000 5 0.017 0.016 0.033 0.017 0.016 0.032 −0.014 −0.013 −0.017 −0.014 −0.013 −0.01710 0.017 0.016 0.032 0.017 0.016 0.032 −0.014 −0.013 −0.017 −0.014 −0.013 −0.01730 0.017 0.015 0.032 0.017 0.016 0.032 −0.014 −0.013 −0.017 −0.014 −0.013 −0.017dist1, Normal distribution for t i ; dist2, distribution with negative kurtosis for t i ; dist3, distribution with positive kurtosis for t i ; dist4, skewed distribution for t i ; dist5,skewed distribution with negative kurtosis for t i ; dist6, skewed distribution with positive kurtosis for t i .On the other hand, a symmetric leptokurtic distribution resultedin a much smaller mean (0.198) and a larger SE (0.290), indicatingthat the center location of the sampling distribution of ˆα wasfurther away from the actual value (0.3) and more uncertaintywas involved in estimating α. With respect to the accuracy of theestimate, Table 2 shows that the normal distribution had a RMSEof 0.251 and a bias value of −0.070. The platykurtic distributiongave rise to smaller but very similar values: 0.244 for RMSE and−0.066 for bias, whereas the leptokurtic distribution had a relativelylarger RMSE value (0.308) and a smaller bias value (−0.102),indicating that it involved more error and negative bias in estimatingα. Hence, under this condition, positive kurtosis affected (thelocation and scale of) the sampling distribution of ˆα as well as theaccuracy of using it to estimate α whereas negative kurtosis didnot. Similar interpretations are used for the 95% interval shownin Table 3, except that one can also use the intervals to determinewww.frontiersin.org February 2012 | Volume 3 | Article 34 | 32