抽樣誤差與點數對MSA估計的影響

Abstract

因素分析為心理及社會科學領域常使用的統計方法之一，在實際進行分析前，研究者常透過檢視Kaiser’s Measure of Sampling Adequacy（MSA，或稱KMO）判斷資料是否適合進行因素分析。然MSA乃是在母群特性下推導的指標，是否會受抽樣誤差影響而產生估計上的偏誤，致使研究者對資料進行因素分析之適切性有錯誤推論，此實係MSA應用於實徵研究上的重要議題。本研究以模擬資料探討在有限樣本中MSA估計值的特性，即其準確性與穩定性。模擬情境涵蓋連續型與次序型資料，操弄的變項包含因素數目、變項因素比、因素負荷量、因素間相關、樣本人數、反應點數、分配型態等。因素數目包含單因素及多因素，每因素的變項數目為3、6與12，因素負荷量為0.4、0.6和0.8，因素間相關包含0相關及0.3、0.5的相關情形，樣本人數為100、200、500與1000人。在次序型資料方面，反應點數有二點、三點、五點與七點，分配型態包含變項偏態為0、1和2，並同步考量皮爾森與多序類相關係數。研究結果顯示，MSA估計偏誤主要受到樣本人數、因素負荷量與變項因素比的影響，樣本人數越少、因素負荷量越低、每因素的變項數目越多，越容易得到低估的MSA估計值，於次序型資料使用皮爾森相關係數所得之MSA估計值的偏誤情形亦較連續型資料嚴重。使用多序類相關係數，在樣本人數大或使用五點量表時會得到較準確的MSA估計值，但於樣本人數少、點數少與變項偏態嚴重時所得之MSA估計值有更多偏誤，其變異程度亦較大，且容易發生無法得到正定矩陣的情況。實徵研究者使用MSA判斷資料是否適合進行因素分析時，宜留意MSA在有限樣本中低估的情況，以避免對資料產生錯誤的推論。

Factor analysis is a frequently used statistical method in psychology and social sciences research. Before applying factor analysis, researchers usually examine whether a data matrix is suitable for factor-analytic methods with Kaiser’s Measure of Sampling Adequacy (MSA, AKA KMO). The MSA measure was developed in the population without considering sampling error. It is therefore worthwhile to investigate the behavior of MSA estimates in finite samples for practical application of this measure. If the estimated MSAs are biased due to sampling errors, misleading inferences are likely to be reached on the factorability of data. This study extended previous research on the evaluation of MSA2 to investigate the accuracy and fluctuations of MSA2 estimates with both continuous and ordinal data. Features manipulated in the present study included the number of factors (M = 1, 3, 5), the number of variables per factor (P/M = 3, 6, 12), factor loading (L = .4, .6, .8), inter-factor correlations (C = 0, .3, .5), and sample sizes (N = 100, 200, 500, 1000). For ordinal variables, we considered the number of response categories (K = 2, 3, 5, 7), the skewness of variables (Sk = 0, 1, 2), as well as the correlation analyzed, Pearson and polychoric correlations. Results indicated that the accuracy of MSA2 estimates was mainly affected by sample sizes, factor loading, and the number of variables per factor. Severe bias of MSA2 occurred when analyzing a large number of weakly correlated variables with insufficient participants sampled. For ordinal data, using Pearson correlations resulted in greater bias in estimation of MSA2 than continuous data. Polychoric correlations might yield more accurate MSA2 estimates with five response categories or large samples. However, larger sample sizes were required to avoid non-Gramian matrices and to obtain stable MSA2 estimates when computing polychoric correlations. In practice, researchers need to bear in mind the downward bias of MSA2 estimates when using this measure as an index for evaluating the factorability of data and interpret the value of sample MSA2s with caution.