終於將我的系統正式搬移到www.apex4u.net的國際網域上了，因此自即日起，我的部落格也要正式搬回家了。

我的系統網址 http://www.apex4u.net

我的部落格網址 http://srl.apex4u.net

歡迎各位參觀。

終於將我的系統正式搬移到www.apex4u.net的國際網域上了，因此自即日起，我的部落格也要正式搬回家了。

我的系統網址 http://www.apex4u.net

我的部落格網址 http://srl.apex4u.net

歡迎各位參觀。

Google 6日推出一款線上試算表程式，雖然仍在進行研發，但是提供一般用戶和企業公司免費替代微軟Excel應用的選擇。 Google的試算表目前仍在測試階段，因此僅限量供應部分用戶。

不過只要有人願意邀請你共同線上編輯同一份試算表，那麼你就可以立即獲得免費的機會。

幸好我已獲得Google寄來的邀請信了，所以有興趣的同學，給我你的Google account，我就可以發邀請信給你測試啦。

不過很遺憾的是，經我實際使用過後，我覺得Google還有很長很長的路要走啊….

字詞搭配(word collocation)是指兩個字習慣性的以同樣形式一起出現在文句中的情形。BBI搭配辭典中如此介紹：

所謂搭配，指的是語詞之間的連用習慣，這是任何語言都有的現象，例如我們中文說「開支票」而不說「寫支票」…英美人士會說 “write a check”，絕不會說 “open a check”…

雖然BBI英語搭配辭典還算方便，但是有的時候還要在那裡翻閱，對於時間寶貴的我而言，實在有點麻煩，所以以直想要找到合適的程式軟體來使用。結果，總算讓我找到相當不錯的網站了。

英國HarperCollins出版公司和伯明罕大學合作的Cobuild計劃，蒐集了數億字的英語母語人士所寫得各種英文文章，成為名曰Bank of English的龐大語料庫，並據以編輯字典等工具書。Cobuild的產品其中有一項為線上查詢英語搭配的網頁。只要用google 找一下cobuild就可以找到啦。好用，太好用了～

When I run my data I get an error message that states that my covariance matrix is not positive definite. I have searched the LISREL book that I have and it doesn't provide any explanation for this error message.

Answer: This message generally means one or more of the following things is happening: 1) There are redundancies among the correlation matrices- in other words, some of the correlations may be a linear function of some of the other correlations. You can fix this by removing the redundant variables or collecting more data. 2) Your model may be estimating more parameters than you have degrees of freedom to use. You can check this by examining how many degrees of freedom you have and the number of parameters you are estimating. The formula for caculating the number of degrees of freedom available to you is q(q+1)/2, where q is the number of measured variables. If you have 9 measured variables, then you must estimate, (9(9+1)/2 = 45), less than 45 parameters. As part of its standard output, LISREL will count the number of parameters it estimates. The difference between the number of estimable parameters and the number of estimated parameters is the number of degrees of freedom used in the chi-square test by LISREL (the first chi-square test to appear on the printout, not the Independence Model chi-square test). 3) LISREL is not correctly reading the raw data, correlation matrix, or covariance matrix. Alternatively, you may be inputting a correlation or covariance matrix which is based on incorrectly read raw data values via PRELIS, SPSS, or another program which has the capability to convert raw data into correlation or covariance matrix form. 4) You computed a covariance or correlation matrix using pairwise deletion of missing data. The solution here is to use a different method of handling missing data.

Not Positive Definite Matrices–Causes and Cures

The seminal work on dealing with not positive definite matrices is Wothke (1993). The chapter is both reabable and comprehensive. This page uses ideas from Wothke, from SEMNET messages, and from my own experience.

The Problem

There are four situations in which a researcher may get a message about a matrix being “not positive definite." The four situations can be very different in terms of their causes and cures.

First, the researcher may get a message saying that the input covariance or correlation matrix being analyzed is “not positive definite." Generalized least squares (GLS) estimation requires that the covariance or correlation matrix analyzed must be positive definite, and maximum likelihood (ML) estimation will also perform poorly in such situations. If the matrix to be analyzed is found to be not positive definite, many programs will simply issue an error message and quit.

Second, the message may refer to the asymptotic covariance matrix. This is not the covariance matrix being analyzed, but rather a weight matrix to be used with asymptotically distribution-free / weighted least squares (ADF/WLS) estimation.

Third, the researcher may get a message saying that its estimate of Sigma (), the model-implied covariance matrix, is not positive definite. LISREL, for example, will simply quit if it issues this message.

Fourth, the program may indicate that some parameter matrix within the model is not positive definite. This attribute is only relevant to parameter matrices that are variance/covariance matrices. In the language of the LISREL program, these include the matrices Theta-delta, Theta-epsilon, Phi () and Psi. Here, however, this “error message" can result from correct specification of the model, so the only problem is convincing the program to stop worrying about it.

“Not Positive Definite"–What Does It Mean?

Strictly speaking, a matrix is “positive definite" if all of its eigenvalues are positive. Eigenvalues are the elements of a vector, e, which results from the decomposition of a square matrix S as:

S = e’Me

To an extent, however, we can discuss positive definiteness in terms of the sign of the “determinant" of the matrix. The determinant is a scalar function of the matrix. In the case of symmetric matrices, such as covariance or correlation matrices, positive definiteness wil only hold if the matrix and every “principal submatrix" has a positive determinant. (“Principal submatrices" are formed by removing row-column pairs from the original symmetric matrix.) A matrix which fails this test is “not positive definite." If the determinant of the matrix is exactly zero, then the matrix is “singular." (Thanks to Mike Neale, Werner Wothke and Mike Miller for refining the details here.)

Why does this matter? Well, for one thing, using GLS estimation methods involves inverting the input matrix. Any text on matrix algebra will show that inverting a matrix involves dividing by the matrix determinant. So if the matrix is singular, then inverting the matrix involves dividing by zero, which is undefined. Using ML estimation involves inverting Sigma, but since the aim to maximize the similarity between the input matrix and Sigma, the prognosis is not good if the input matrix is not positive definite. Now, some programs include the option of proceeding with analysis even if the input matrix is not positive definite–with Amos, for example, this is done by invoking the $nonpositive command–but it is unwise to proceed without an understanding of the reason why the matrix is not positive definite. If the problem relates to the asymptotic weight matrix, the researcher may not be able to proceed with ADF/WLS estimation, unless the problem can be resolved.

In addition, one interpretation of the determinant of a covariance or correlation matrix is as a measure of “generalized variance." Since negative variances are undefined, and since zero variances apply only to constants, it is troubling when a covariance or correlation matrix fails to have a positive determinant.

Another reason to care comes from mathematical statistics. Sample covariance matrices are supposed to be positive definite. For that matter, so should Pearson and polychoric correlation matrices. That is because the population matrices they are supposedly approximating *are* positive definite, except under certain conditions. So the failure of a matrix to be positive definite may indicate a problem with the input matrix.

Why is My Matrix Not Positive Definite, and What Can I Do About It?

Properly, the question is, why does the matrix contain zero or negative eigenvalues. However, it may be easier for many researchers to think about why the determinant is zero or negative? Either way, there are many possibilities, and there are different possible solutions that go with each possible cause.

Further, there are other solutions which sidestep the problem without really addressing its cause. These options carry potentially steep cost. They are discussed separately, below.

Linear Dependency

A not positive definite input covariance matrix may signal a perfect linear dependency of one variable on another. For example, if a plant researcher had data on corn (maize) stalks, and two of the variables in the covariance matrix were “plant height" and “plant weight," the linear correlation between the two would be nearly perfect, and the covariance matrix would be not positive definite within sampling error. It may be easier to detect such relationships by sight in a correlation matrix rather than a covariance matrix, but often these relationships are logically obvious. Multivariate dependencies, where several variables together perfectly predict another variable, may not be visually obvious. In those cases, sequential analysis of the covariance matrix, adding one variable at a time and computing the determinant, should help to isolate the problem. (I would use a spreadsheet program for this, like Microsoft (TM) Excel (TM), for convenience.)

Dealing with this kind of problem involves changing the set of variables included in the covariance matrix. If two variables are perfectly correlated with each other, then one may be deleted. Alternatively, principal components may be used to replace a set of collinear variables with one or more orthogonal components.

In regard to the asymptotic weight matrix, the linear dependency exists not between variables, but between elements of the moments (the means and variances and covariances or the correlations) which are being analyzed. This can occur in connection with modeling multiplicative interaction relationships between latent variables. Jöreskog and Yang (1996) show how moments of the interaction construct are linear functions of moments of the “main effect" constructs. Their article explores alternative approaches for estimating these models

Error Reading the Data

If the problem is with your input matrix in particular, first make sure that the program has read your data correctly. Remember, an empty covariance matrix (with no variables in it) is always not positive definite. Try reading the data using another program, which will allow you to validate the covariance matrix estimated by the SEM program. If you generated the covariance matrix with one program, and are analyzing it with another, make sure that the covariance matrix was read correctly. This can be particularly problematic when the asymptotic weight matrix is the focus of the problem.

Typographical Error

Whenever a covariance matrix is transcribed, there is a chance of error. So if you just have the matrix (say, from a published article, but not the data itself, double-check for transcription errors. Also remember that journals are not perfect, so a covariance matrix in an article may also contain an error. In a recent case, for example, it appeared that the sign of a single (relatively large) coefficient was reversed at some point, and this reversal made the matrix not positive definite. In that case, changing the sign of that one coefficient eliminated the problem.

Starting Values

The model-implied matrix Sigma is computed from the model’s parameter estimates. Especially before iterations begin, those estimates may be such that Sigma is not positive definite. So if the problem relates to Sigma, first make sure that the model has been specified correctly, with no syntax errors. If the proposed model is “unusual," then the starting value routines that are incorporated into most SEM programs may fail. Then it is up to the researcher to supply likely starting values.

Sampling Variation

When sample size is small, a sample covariance or correlation matrix may be not positive definite due to mere sampling fluctuation. As most matrices rapidly converge on the population matrix, however, this in itself is unlikely to be a problem. Anderson and Gerbing (1984) documented how parameter matrices (Theta-Delta, Theta-Epsilon, Psi and possibly Phi) may be not positive definite through mere sampling fluctation. Most often, such cases involve “improper solutions," where some variance parameters are estimated as negative. In such cases, Gerbing and Anderson (1987) suggested that the offending estimates could be fixed to zero with minimal harm to the program.

Estimators of the asymptotic weight matrix converge much more slowly, so problems due to sampling variation can occur at much larger sample sizes (Muthén & Kaplan, 1985, 1992). Using an asymptotic weight matrix with polychoric correlations appears to compound the problem. Where sampling variation is the issue, Yung and Bentler (1994) have proposed a bootstrapping approach to estimating the asymptotic weight matrix, which may avoid the problem.

Missing Data

Large amounts of missing data can lead to a covariance or correlation matrix not positive definite. With simple replacement schemes, the replacement value may be at fault. With pairwise deletion, the problem may arise precisely because each element of the covariance matrix is computed from a different subset of the cases (Arbuckle, 1996). To check whether this is the cause, use a different missing data technique, such as a different replacement value, listswise deletion or (perhaps ideally) a maximum likelihood/EMCOV simultaneous estimation method.

My Variable is a Constant!

Sometimes, either through an error reading data or through the process of deleting cases that include missing data, it happens that some variable in a data set takes on only a single value. In other words, one of the variables is actually a constant. This variable will then have zero variance, and the covariance matrix will be not positive definite. Simple tabulation of the data will provide a forewarning of this. If this is the problem, either the researcher must choose a different missing-data strategy, or else the variable must be deleted.

Polychoric Correlations

Programs that estimate polychoric correlations on a pairwise basis–one correlation at a time–may yield input correlation matrices that are not positive definite. Here the problem occurs because the whole correlation matrix is not estimated simultaneously. It appears that this is most likely to be a problem when the correlation matrix contains large numbers of variables. Try computing a matrix of Pearson correlations and see whether the problem persists.

If the problem lies with the polychoric correlations, there may not be a good solution. One approach is to use a program, like EQS, that includes the option of deriving all polychoric correlations simultaneously, rather than one at a time (cf., Lee, Poon & Bentler, 1992). But be warned–Joop Hox reports that the computational burden is enormous, and it increases exponentially with the number of variables.

Ed Cook has experimented with an eigenvalue/eigenvector decomposition approach. If a covariance or correlation matrix is not positive definite, then one or more of its eigenvalues will be negative. After decomposing the correlation matrix into eigenvalues and eigenvectors, Ed Cook replaced the negative eigenvalues with small (.05) positive values, used the new values to compute a covariance matrix, then standardized the resulting matrix (diving by the square root of the diagonal values) so that the result was again was a correlation matrix. Ed reported that the bias resulting from this process appeared to be small.

No Error Variance

Sometimes researchers specify zero elements on the diagonals of Theta-delta or Theta-epsilon. A zero here implies no measurement error. While it may seem unlikely, on reflection, that any latent variable could be measured without error, nevertheless the practice is common, when a construct has only a single measure. Single measures often lead to identification problems, and analysts may leave the parameter fixed at zero by default. If a diagonal element is fixed to zero, then the matrix will be not positive definite. However, since this is precisely what the researcher intended to do, there is no cause for alarm. The only problem is that these values may cause the solution to fail an “admissibility check," which may lead to premature termination of the iterative estimation process. In such cases, it is merely a matter of disabling the admissibility check. In LISREL, for example, this is done by adding AD=OFF to the OUtput line.

Negative Error Variance

Negative values on the diagonal are another matter. Since the diagonal elements of these matrices are variance terms, negative values are unacceptable. Further, since these error variances represent the “left-over" part of some variable, a negative error variance suggests that the regression has somehow explained more than 100 percent of the variance. In my own experience, these values are symptoms of a serious fit problem. Comprehensive fit assessment will help the researcher to isolate the specific problem.

Sidestepping the Problem

As with many problems, there are ways to sidestep this problem without actually trying to discern its cause. Besides simply compelling the program to proceed with its analysis, researchers can make a ridge adjustment to the covariance or correlation matrix. This involves adding some quantity to the diagonal elements of the matrix. This addition has the effect of attenuating the estimated relations between variables. A large enough addition is sure to result in a positive definite matrix. The price of this adjustment, however, is bias in the parameter estimates, standard errors, and fit indices. Partial least squares methods may also proceed with no regard for the determinant of the matrix, but this involves an entirely different methodology.

References

Anderson, J. C., & Gerbing, D. W. (1984). The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analysis. Psychometrika, 49(2–June), 155-73.

Arbuckle, J. L. (1996). Full information estimation in the presence of incomplete data. In G. A. Marcoulides & R. E. Schumacker (Eds.), Advanced structural equation modeling: Issues and techniques (pp. 243-78). Mahwah, NJ: Lawrence Erlbaum.

Gerbing, D. W., & Anderson, J. C. (1987). Improper solutions in the analysis of covariance structures: Their interpretability and a comparison of alternate respecifications. Psychometrika, 52(1–March), 99-111.

Jöreskog, K. G., & Yang F. [now Fan Yang Jonsson] (1996). Nonlinear structural equation models: The Kenny-Judd model with interaction effects. In G. A. Marcoulides & R. E. Schumacker (Eds.), Advanced structural equation modeling: Issues and techniques (pp. 57-88). Mahwah, NJ: Lawrence Erlbaum.

Lee, S.-Y., Poon, W.-Y., & Bentler, P. M. (1992). Structural equation models with continuous and polytomous variables. Psychometrika, 57(1–March), 89-105.

Muthén, B. & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables. British Journal of Mathematical and Statistical Psychology, 38, 171-89.

Muthén, B. & Kaplan, D. (1992). A comparison of some methodologies for the factor analysis of non-normal Likert variables: A note on the size of the model. British Journal of Mathematical and Statistical Psychology, 45, 19-30.

Wothke, W. (1993). Nonpositive definite matrices in structural modeling. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 256-93). Newbury Park, CA: Sage.

Yung, Y.-F., & Bentler, P. M. (1994). Bootstrap-corrected ADF test statistics in covariance structure analysis. British Journal of Mathematical and Statistical Psychology, 47, 63-84.

——————————————————————————–

http://www.gsu.edu/~mkteer/npdmatri.html

Return to the SEMNET FAQ home page.

Return to Ed Rigdon’s home page.

Last updated: June 11, 1997

我們看見醫院裡的開刀房，醫生、護士總喜歡穿上綠色的衣服，這是為什麼？原

來是因為大腦對紅綠顏色辨識出了問題的關係。

很有意思！好神奇哦，試試看！自己的眼睛也不一定值得相信！

如果你的眼睛順著粉紅色的點移動，你只能看到一種顏色：粉紅。如果你盯著中

間的黑色的"+"，運動的點將會變成綠色。如果繼續盯住黑色的"+" 過一會兒，所有的

粉紅色慢慢全部消失，你將只會看到一個綠色的點在運動。

這就是你大腦工作的方式，其實圖中根本就不存在綠色的點，粉紅色的點根本沒

有消失。

大家應該知道了吧？即使親眼所見，也不一定是真的！

LISREL模式中包含4種變項：潛在自變項（ξ：外生變數）、潛在依變項（η：內生變數）、潛在自變項的觀察變項（X）、潛在依變項的觀察變項（Y） 。

**learning model**** **

**observed variables **

**x1 x2 x3 x4 y1 y2 y3 y4 y5 y6 y7**** **

**correlation matrix **

** ****1**** **

** 0.56 1**** **

** 0.41 0.42 1**** **

** 0.43 0.38 0.55 1**** **

** 0.44 0.47 0.41 0.38 1**** **

** 0.48 0.45 0.45 0.41 0.52 1**** **

** 0.41 0.40 0.22 0.26 0.28 0.21 1**** **

** 0.43 0.39 0.23 0.25 0.25 0.24 0.55 1**** **

** 0.39 0.38 0.41 0.36 0.29 0.27 0.39 0.40 1**** **

** 0.36 0.39 0.36 0.44 0.33 0.27 0.35 0.37 0.51 1**** **

** 0.30 0.42 0.45 0.31 0.29 0.32 0.34 0.31 0.27 0.39 1**** **

** 0.31 0.41 0.44 0.36 0.22 0.24 0.39 0.23 0.28 0.37 0.58 1**** **

** ****sample size****＝****199**

**latent variables**** **

** a1 b1 b2 b3**** **

** relationships**** **

** ****x1 x2 x3 x4=a1**** **

** y1 y2 y3=b1**** **

** y4 =b2**** **

** y5 y6 y7=b3**** **

** b1=a1 b2**** **

** b2=a1**** **

** b3=a1 b1 b2 **

** ****set the errors of y4 to 0 **

** ****print residuals**** **

** path diagram**** **

** end of problem**

模式適配度

指標 | 理想數值 |

χ2值比率 | 1~3之間 |

GFI | 0.9(含)以上 |

AGFI | 0.9 (含)以上 |

RMR | 至少小於0.1 |

RMSEA | 0.05~0.08 |

NFI | 0.9 (含)以上 |

IFI | 0.9 (含)以上 |

NNFI | 0.9 (含)以上 |

SketchUp是一套繪製3D圖形的軟體，儘管他不是一相當專業的3D繪圖軟體，但是親近且直覺的操作介面與環境，相當適合那些想要嘗試學習3D的人使用，而且他的操作方法實在是相當簡單，我想國小的學生都可以運用得非常好。

因為他的優點真是不少，所以現在Google已經將他買下來，並且提供個人用戶免費使用喔！！原價一套記得要幾千元呢，不過如果作為商業用途就要收費就是了。

這相當好的軟體，如果想學3D電腦影像，這會是一套相當好用的軟體，如果想要瞭解幾何圖像，這也會是相當好的輔助工具。

天氣陰濛濛的，就像我的心情一樣，完全看不到任何色彩。

這陣子以來，雖然稱不上是艱難的時刻，但許多的事情卻讓我感到格外的無力，就像經過一場長時間的馬拉松競賽一樣。其實一直都很想找個時間、找個機會坐下來，靜靜的什麼也不動，什麼也不想，感受時間流逝的無聲、無息。

又有的時候，我猛然從沉靜已久的思索中回神，走了那麼大的一段路，曾經所走過的每一步，如今都像是用盡我的氣力般，深刻地將足跡烙在我的記憶裡…，我像是要擺脫夜巷中傳來的腳步聲，無從選擇地加緊步伐，只希望不要再聽到這惱人的鬼影魅聲。

這個時代的程式設計師還挺好命的，寫程式越來越容易，越來越方便，一大堆現有的API可用不說，現在還有許多開發好的程式套件，直接呼叫就可以跑了，呵呵，回想我當初用Assembly寫程式的時候，可沒那麼好命啊…真好，現在當紅的Skeype也有套件可用了，可以和.net兜在一起，寫個程式將Skeype整合到自己的系統來，變得很方便了呢，畢竟現在用語音通訊才符合人性啊。所以可預期的將來，大部分的線上溝通會以語音為主，同時加上即時錄音的功能，線上會議記錄也很方便，不過語音終究有其限制，我想要成為正式會議的資料，還時非得用文件形式不可，除非語音辨識的水準能趕得上，不過以現在IBM viaVoice來看，我想還需要十年吧，哈，也許不止總之，感謝科技的進步，套一句我老師曾說過的…”現在還有誰不會寫程式嗎？”