Tuesday, March 31, 2009
Cronbach’s Alpha Rule of Thumb
To examine reliability and internal consistency Cronbach’s alpha tests were conducted using the survey subscales: communication, organizational commitment, organizational justice, organizational citizenship, affect based trust, cognition based trust and resistance to change. George and Mallery (2003) suggest the following rules of thumb for evaluating alpha coefficients, “> .9 excellent, > .8 good, > .7 acceptable, > .6 questionable, > .5 poor, < .5 unacceptable.” The measure (alpha or α) assumes that items measuring the same thing will be highly correlated (Welch & Comer, 1988). Click here for dissertation statistics help
Number of Components to Extract
There are a number of ways to determine the number of factors to extract. The Kaiser criterion suggests that one should retain any factors with eigenvalues greater than one. Scanning the Total Variance Explained (see Appendix C), the first eleven components (λ or eigenvalues) are above one. The total variance explained for the first two components is 27.683 and 7.598 percent of the variance, respectively. The total variance explained for the first two factors is 5.281 percent. Total variance explained by the first eleven factors is 72.012 percent. The Kaiser technique is only reliable, however, when the number of variables is less than thirty and the communalities are greater than .7. Inspection of the communalities shows twenty three of the 54 communality coefficients are greater than .7. Communalities (h2) “represent how much of the variance of a measured variable was useful in delineating the extracted factors” (Thompson, 2004, p. 61). These communalities also represent the R-square (R2) between the factor scores (latent variable scores) and measured scores on the measured variable.
The Catell (1996) technique suggests keeping factors above the elbow in the Scree plot (see Figure 1 in Appendix B). In this study, the scree plot suggests two factors be retained. Breaks also appear for 3, 4, 5, and 7 components. The seven factor solution supports the theoretical model with seven factors. When this model was specified, however, the majority of the variables loaded onto the first factor and the model made no theoretical sense. Theoretical considerations supported a nine factor model. The pattern matrix is shown in Table 4.
The items cluster cleanly into the communication, commitment, citizenship, cognition based trust, and resistance to change factors. Nearly all the affect based trust measures hang together with the exception of abt6 (This person approaches his/her job with professionalism and dedication. (ABT-6)) which is strongly correlated with the items that measure cognition based trust. The variable, organizational justice, includes three subfactors, fairness, employee voice and justification. The first four items for fairness cluster together and appear to measure that concept. The questions are shown below. Survey questions Orgjv6, Orgjv7, Orgjv8 and OrgjJ11 cluster together to form the Employee Voice measure. Similarly, Orgjv5, Orgjv8, and OrgjJ10 load onto Justification. These subscales will be utilized as one scale, organization justice, in the computation of Cronbach’s alpha.
Click here for dissertation statistics help
The Catell (1996) technique suggests keeping factors above the elbow in the Scree plot (see Figure 1 in Appendix B). In this study, the scree plot suggests two factors be retained. Breaks also appear for 3, 4, 5, and 7 components. The seven factor solution supports the theoretical model with seven factors. When this model was specified, however, the majority of the variables loaded onto the first factor and the model made no theoretical sense. Theoretical considerations supported a nine factor model. The pattern matrix is shown in Table 4.
The items cluster cleanly into the communication, commitment, citizenship, cognition based trust, and resistance to change factors. Nearly all the affect based trust measures hang together with the exception of abt6 (This person approaches his/her job with professionalism and dedication. (ABT-6)) which is strongly correlated with the items that measure cognition based trust. The variable, organizational justice, includes three subfactors, fairness, employee voice and justification. The first four items for fairness cluster together and appear to measure that concept. The questions are shown below. Survey questions Orgjv6, Orgjv7, Orgjv8 and OrgjJ11 cluster together to form the Employee Voice measure. Similarly, Orgjv5, Orgjv8, and OrgjJ10 load onto Justification. These subscales will be utilized as one scale, organization justice, in the computation of Cronbach’s alpha.
Click here for dissertation statistics help
Univariate Outliers
Outliers can cause serious problems for any regression-based test such as SEM. Due to distance separation from the normal data swarm outliers tend to make the regression line deviate in the direction of the outlier. Outliers can appear in both univariate and multivariate situations but Tabachnick and Fidell (2001) suggest first assessing univariate outliers. Outliers can be assessed using bivariate scatterplots or boxplots. Examination of the boxplots revealed some variables have outliers that are between 1.5 and three box lengths from the top or bottom edge of the box. The boxplots are in Appendix A. The data were reexamined to check for accuracy. All values were within the expected range and appeared to be a legitimate part of the sample so will remain in the analysis.
Factor analysis has also been articulated by numerous seminal authors as a vital component of a model’s validity. Churchill (1979) stated, “Factor analysis can then be used to confirm whether the number of dimensions can be verified empirically” (p. 69). Nunnally (1978) also stated, “factor analysis is intimately involved with questions of validity…Factor analysis is at the heart of the measurement of psychological constructs” (pp. 112-113). The factor-analytic stage (EFA) therefore, is an assessment of one aspect of model validity.
Principle components analysis transforms the original measures into a smaller set of linear combinations with all of the variance being used (Pallant, 2003). Factor analysis (FA) is similar; however it uses a mathematical model and only analyzes shared variance. Tabachnick and Fidell (2001) described the difference between EFA and FA: “If you are interested in a theoretical solution uncontaminated by unique and error variability, FA is your choice. If on the other hand you want an empirical summary of he data set, PCA is the better choice” (p. 611). In addition, PCA yields components whereas FA provides factors. Sometimes they are used interchangeably. This study will follow the factor analysis guidelines of Mertler and Vannatta (2005).
There are two basic requirements to factor analysis: sample size and the strength of the relationship of the measures. The sample size of 286 is close to the 300 recommended by Tabachnick and Fidell (2001) and is sufficient. The authors also caution that a matrix that is factorable should include correlations in excess of .30. If none are found, reconsider use of factor analysis.
An inspection of the correlation matrix in Table 3 shows less than half the values < .3 as recommended by Tabachnick and Fidell (2001). Principal axis factoring with Promax rotation was performed to see if the items that were written for the survey to index the seven constructs (communication, commitment, organizational justice, organizational citizenship, affect based trust, cognition based trust and resistance to change) actually do hang together. That is, are the participants’ responses to the communication questions more similar to each other than their responses to the commitment items? The Kaiser-Meyer-Olkin (Kaiser, 1970, 1974) Measure of Sampling Adequacy (KMO) value of .891 exceeds the recommended value of .6 and the Bartlett’s Test of Sphericity (Bartlett, 1954) reached statistical significance (Chi-square = 11,880.86 (p < .001; df = 1431)) supporting the factorability of the correlation matrix. Examination of the correlation matrix revealed some collinearity (Determinant = 0.000) between variable pairs. In example, the correlation between cmit6 and comu4 was .72. The first analysis resulted in eleven factors, however, these made no theoretical sense, but were based on Eigenvalues greater than one. For instance, factor 11 included no variables. These results are shown in Appendix B. To determine which questions load on a factor, the cutoff of .278 was chosen (twice the significant correlation of a sample of 350 at the .01 level). However, when the question loading magnitude was much greater on another factor, the question was identified loading just on that factor. The variables from the following questions have been reverse coded: Organizational Justice (ORGJ-1), (ORG-2), (ORG-6), (ORG-7) (ORG-9), and (ORG-11). The variables from the following questions have been reverse coded: Resistance to Change (RCHG-2) and (RCHG-6). Click here for dissertation statistics help
Factor analysis has also been articulated by numerous seminal authors as a vital component of a model’s validity. Churchill (1979) stated, “Factor analysis can then be used to confirm whether the number of dimensions can be verified empirically” (p. 69). Nunnally (1978) also stated, “factor analysis is intimately involved with questions of validity…Factor analysis is at the heart of the measurement of psychological constructs” (pp. 112-113). The factor-analytic stage (EFA) therefore, is an assessment of one aspect of model validity.
Principle components analysis transforms the original measures into a smaller set of linear combinations with all of the variance being used (Pallant, 2003). Factor analysis (FA) is similar; however it uses a mathematical model and only analyzes shared variance. Tabachnick and Fidell (2001) described the difference between EFA and FA: “If you are interested in a theoretical solution uncontaminated by unique and error variability, FA is your choice. If on the other hand you want an empirical summary of he data set, PCA is the better choice” (p. 611). In addition, PCA yields components whereas FA provides factors. Sometimes they are used interchangeably. This study will follow the factor analysis guidelines of Mertler and Vannatta (2005).
There are two basic requirements to factor analysis: sample size and the strength of the relationship of the measures. The sample size of 286 is close to the 300 recommended by Tabachnick and Fidell (2001) and is sufficient. The authors also caution that a matrix that is factorable should include correlations in excess of .30. If none are found, reconsider use of factor analysis.
An inspection of the correlation matrix in Table 3 shows less than half the values < .3 as recommended by Tabachnick and Fidell (2001). Principal axis factoring with Promax rotation was performed to see if the items that were written for the survey to index the seven constructs (communication, commitment, organizational justice, organizational citizenship, affect based trust, cognition based trust and resistance to change) actually do hang together. That is, are the participants’ responses to the communication questions more similar to each other than their responses to the commitment items? The Kaiser-Meyer-Olkin (Kaiser, 1970, 1974) Measure of Sampling Adequacy (KMO) value of .891 exceeds the recommended value of .6 and the Bartlett’s Test of Sphericity (Bartlett, 1954) reached statistical significance (Chi-square = 11,880.86 (p < .001; df = 1431)) supporting the factorability of the correlation matrix. Examination of the correlation matrix revealed some collinearity (Determinant = 0.000) between variable pairs. In example, the correlation between cmit6 and comu4 was .72. The first analysis resulted in eleven factors, however, these made no theoretical sense, but were based on Eigenvalues greater than one. For instance, factor 11 included no variables. These results are shown in Appendix B. To determine which questions load on a factor, the cutoff of .278 was chosen (twice the significant correlation of a sample of 350 at the .01 level). However, when the question loading magnitude was much greater on another factor, the question was identified loading just on that factor. The variables from the following questions have been reverse coded: Organizational Justice (ORGJ-1), (ORG-2), (ORG-6), (ORG-7) (ORG-9), and (ORG-11). The variables from the following questions have been reverse coded: Resistance to Change (RCHG-2) and (RCHG-6). Click here for dissertation statistics help
MVA Analysis
The SPSS missing value analysis (MVA) was used to analyze the data for both MAR and MCAR data loss using an expectation maximization technique. Little’s (Little & Rubin, 2002) MCAR resulted in a Chi-square = 1852.25 (p = 0.099; df = 1778). This significance denotes that the missing data is MCAR and the data loss pattern is not systematic.
The SPSS MVA module also incorporates an expectation-maximization (EM) algorithm for generation of imputed values used to fill in all the missing data. Since the data is MCAR, listwise deletion is a better alternative than pairwise deletion which may cause covariance matrix issues due to unequal numbers of cases (Kline, 2005).
The AMOS application is unique in that it can be used to analyze data that includes missing data. AMOS incorporates a special form of maximum likelihood estimation (Special ML) which partitions all cases with the same missing data patterns. Peters and Enders (2002) found that this method for analyzing datasets with incomplete data “outperformed traditional (available case) methods” (cited in Kline, 2005, p. 56). Tabachnick and Fidell (2001) suggest using both methods (with and without missing data) but favor the EM imputation method and listwise methods (if data is ignorable) over mean substitution or pairwise deletion. Tabachnick and Fidell (2001) state, “The decision about to handle missing data is important. At best, the decision is among several bad alternatives” (p. 59).
Caution should be exercised with any method using a dataset with a high percentage of missing values (> 5%). Nunnally and Bernstein (1994) suggest that when there is a high percentage of missing values any of these methods may be unsatisfactory. Incorporating listwise deletion may be the best option for MCAR data since EM imputation may cause distorted coefficients of association and correlations (Kalton & Kasprzyk, 1982). In the present data set, listwise deletion resulted in a final sample size of 286 respondents.
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The SPSS MVA module also incorporates an expectation-maximization (EM) algorithm for generation of imputed values used to fill in all the missing data. Since the data is MCAR, listwise deletion is a better alternative than pairwise deletion which may cause covariance matrix issues due to unequal numbers of cases (Kline, 2005).
The AMOS application is unique in that it can be used to analyze data that includes missing data. AMOS incorporates a special form of maximum likelihood estimation (Special ML) which partitions all cases with the same missing data patterns. Peters and Enders (2002) found that this method for analyzing datasets with incomplete data “outperformed traditional (available case) methods” (cited in Kline, 2005, p. 56). Tabachnick and Fidell (2001) suggest using both methods (with and without missing data) but favor the EM imputation method and listwise methods (if data is ignorable) over mean substitution or pairwise deletion. Tabachnick and Fidell (2001) state, “The decision about to handle missing data is important. At best, the decision is among several bad alternatives” (p. 59).
Caution should be exercised with any method using a dataset with a high percentage of missing values (> 5%). Nunnally and Bernstein (1994) suggest that when there is a high percentage of missing values any of these methods may be unsatisfactory. Incorporating listwise deletion may be the best option for MCAR data since EM imputation may cause distorted coefficients of association and correlations (Kalton & Kasprzyk, 1982). In the present data set, listwise deletion resulted in a final sample size of 286 respondents.
Click here for dissertation statistics help
MIssing Values
According to Tabachnick and Fidell (2001), “Missing data is one of the most pervasive problems in data analysis “ (p. 58). Missing data can have serious effects on the reliability, validity and generalizability of the data (Tabachnick & Fidell, 2001). Missing data can be indicative of lack of knowledge, fatigue or sensitivity, or interpretation by the respondent of the questionnaire relevance. When the number of missing cases is small (< 5%) it is common to exclude the cases from the analysis (Tabachnick & Fidell, 2001). In the present analysis, every variable is missing at least 16% of the responses. The univariate statistics are shown below in Table 2.
Before exploratory factor analysis it must be determined if missing data is systematic (represents bias) or is ignorable. Missing data also has other important ramifications, especially in factor analysis. Factor analysis using listwise deletion should not be conducted unless the missing data is at least missing completely at random (MCAR).
Before exploratory factor analysis it must be determined if missing data is systematic (represents bias) or is ignorable. Missing data also has other important ramifications, especially in factor analysis. Factor analysis using listwise deletion should not be conducted unless the missing data is at least missing completely at random (MCAR).
Normality
Data normality is focused on the premise that data is from one or more normally distributed populations. Characteristics of a distribution can be described by its moments (the average of its values which are then raised to a certain power). The normal distribution in standard form has a first moment (mean) of zero, a second moment (variance) of one, a third moment (skewness) of zero and a fourth moment (kurtosis) of three. Many statistical programs like SPSS subtract the three from the kurtosis value to normalize to zero for reporting purposes. These statistics are based on the distribution curve as a whole and not on individual cases. Data normality is usually focused on skewness and kurtosis which are measures of shape. A skewness and kurtosis of zero is indicative of a normal distribution. Skewness is associated with the symmetry of the distribution. Kurtosis is associated with how peaked or flat the distribution is. A kurtosis above zero is indicative of a peaked distribution while a negative value is indicative of a flat distribution. Some authors suggest that univariate values approaching at least 2.0 for skewness and 7.0 for kurtosis should be suspect (West et al., 1995; Yuan & Bentler, 1999). The descriptive statistics, including skewness and kurtosis are shown below in Table 1. Examination of the distributions indicated only one variable, cmit8 has a high negative skew, -2.179. Computing the log transformation reduced the skew to .851 and the kurtosis to.247. The transformed variable will be used in further analyses.
Screening of the Data
Careful analysis of data applicability after collection and before analysis is probably the most time-consuming part of data analysis (Tabachnick & Fidell, 2001). This step is, however, of utmost importance as it provides the foundation for any subsequent analysis and decision-making which rests on the accuracy of the data. Incorrect analysis of the data during purification, including EFA, and before conducting confirmatory SEM analysis may result in poor fitting models or, worse, models that are inadmissible.
Data screening is important when employing covariance-based techniques such as structural equation modelling where assumptions are stricter than for the standard t-test. Many of the parametric statistical tests (based on probability distribution theory) involved in this study assume that: (a) normally distributed data – the data are from a normally distributed population, (b) homogeneity of variance – the variances in correlational designs should be the same for each level of each variable, (c) interval data – data where the distance between any two points is the same and is assumed in this study for Likert data, and (d) independence – the data from each respondent has no effect on any other respondent’s scores.
Many of the common estimation methods in SEM (such as maximum-likelihood estimation) assume: (a) “all univariate distributions are normal, (b) joint distribution of any pair of the variables is bivariate normal, and (c) all bivariate scatterplots are linear and homoscedastic” (Kline, 2005, p. 49). Unfortunately, SPSS does not offer an assessment of multivariate normality but Field (2005) and others (Kline, 2005; Tabachnick & Fidell, 2001) recommend first assessing univariate normality. The data were checked for plausible ranges and examination was satisfactory. There were no data out of range.
Data screening is important when employing covariance-based techniques such as structural equation modelling where assumptions are stricter than for the standard t-test. Many of the parametric statistical tests (based on probability distribution theory) involved in this study assume that: (a) normally distributed data – the data are from a normally distributed population, (b) homogeneity of variance – the variances in correlational designs should be the same for each level of each variable, (c) interval data – data where the distance between any two points is the same and is assumed in this study for Likert data, and (d) independence – the data from each respondent has no effect on any other respondent’s scores.
Many of the common estimation methods in SEM (such as maximum-likelihood estimation) assume: (a) “all univariate distributions are normal, (b) joint distribution of any pair of the variables is bivariate normal, and (c) all bivariate scatterplots are linear and homoscedastic” (Kline, 2005, p. 49). Unfortunately, SPSS does not offer an assessment of multivariate normality but Field (2005) and others (Kline, 2005; Tabachnick & Fidell, 2001) recommend first assessing univariate normality. The data were checked for plausible ranges and examination was satisfactory. There were no data out of range.
Assessing Reliability and Validity of Constructs and Indicators
One of the most important advantages offered by latent-variable analyses is the opportunity that is provided to assess the reliability and validity of the study’s variables. In general, reliability refers to consistency of measurement; validity refers to the extent to which an instrument measures what it is intended to measure. For example, a survey is reliable if it provides essentially the same set of responses for a group of respondents upon repeated administration. Similarly, if a scale is developed to measure marketing effectiveness and scores on the scale do in fact reflect respondents’ underlying levels of marketing performance, then the scale is valid. For both reliability and validity, there are a number of different ways that they may be measured.
Indicator reliability. The reliability of an indicator (observed variable) is defined as the square of the correlation (squared multiple correlation or SMC) between a latent factor and that indicator. For instance, looking at Table 1, the standardized loading for the path between Sympathique and F1 is 0.970 and the reliability is 0.939. Looking at the range of indicator reliabilities, many have relative high reliabilities (0.6 and above), however, several have really low reliabilities, like Effacee with an indicator reliability of 0.313.
Composite reliability has been computed for each latent factor included in the model. This index is similar to coefficient and reflects the internal consistency of the indicators measuring a particular factor (Fornell and Larcker, 1981). Both the composite reliability and the variance extracted estimates are shown in Table 1. Fornell and Larcker (1981) recommend a minimum composite reliability of .60. An examination of the composite reliabilities revealed that all meet that minimum acceptable level.
The variance extracted estimates assesses the amount of variance that is explained by an underlying factor in relation to the amount of variance due to measurement error. For instance, the variance estimate for F1 was 0.838, meaning that 83.8% of the variance is explained by the F1 construct, and 16.2% is due to measurement error. Fornell and Larcker (1981) suggest that constructs should exhibit estimates of .50 or larger. Estimates less than .50 indicate that variance due to measurement error is larger than the variance captured by the factor. The variance extracted estimates all meet this minimum threshold, so the validity of the latent construct as well as the associated constructs is acceptable. It should also be noted that Hatcher (1994), cautions that the variances extracted estimate test is conservative; reliabilities can be acceptable even if variances extracted estimates are less than .50.
Convergent validity is present when different instruments are used to measure the same construct and scores from these different instruments are strongly correlated. In contrast, discriminant validity is present when different instruments are used to measure different constructs and the measures of these different constructs are weakly correlated.
In the present study, convergent validity was assessed by reviewing the t-tests for the factor loadings. If all the factor loadings for the indicators were greater than twice their standard errors, the parameter estimates demonstrated convergent validity. That all t-tests are significant showed that all indicators were effectively measuring the same construct (Anderson & Gerbing, 1988). Consider the convergent validity of the ten indicators that measure F1. The results show that the t-values for these ten indicators range from -14.480 to 18.510. These results support the convergent validity of Sympathique, Desagreable, Amicale, Souple, Severe, Autoritaire, Compatissante, au coeur tender, Spontanée, Distante, and Attentive aux autres as measures of F1.
Discriminant validity was assessed through the use of variance extracted test. Constructs were evaluated by comparing the variance extracted estimates for two factors, and then compared with the square of the correlation between the two factors. Discriminant validity is demonstrated if both variance extracted estimates are greater than the squared correlation. In the present study, the correlation between the factors F1 and F2 was 0.154; the squared correlation was 0.024. The correlations and squared correlations are shown in Table 2. The variance extracted estimate was 0.838 for F1 and 0.666 for F2. Because the variance extracted estimates are greater than the square of the interfactor correlation, the test supports the discriminant validity of these two factors. Examination of the other variance extracted estimates and squared correlation coefficients supported discriminant validity within the model.
References
Anderson, J.C. & Gerbing, D.W. (1988). Structural equation modeling in practice: A
review and recommended two-step approach. Psychological Bulletin, 103, 411-423.
Bollen, K.A. (1989). Structural equations with latent variables. New York: John Wiley
& Sons.
Fornell, C. & Larcker, D.F. (1981). Evaluating structural equation models with
unobservable variables and measurement error. Journal of Marketing Research, 18,
39-50.
Hatcher, L. (1994) A step-by-step approach to using SAS for factor analysis and
structural equation modeling. Cary, NC: SAS Institute Inc.
Jöreskog, K.G. & Sörbom, D. (1989). LISREL 7: A guide to the program and
application, 2nd edition. Chicago: SPSS Inc.
Indicator reliability. The reliability of an indicator (observed variable) is defined as the square of the correlation (squared multiple correlation or SMC) between a latent factor and that indicator. For instance, looking at Table 1, the standardized loading for the path between Sympathique and F1 is 0.970 and the reliability is 0.939. Looking at the range of indicator reliabilities, many have relative high reliabilities (0.6 and above), however, several have really low reliabilities, like Effacee with an indicator reliability of 0.313.
Composite reliability has been computed for each latent factor included in the model. This index is similar to coefficient and reflects the internal consistency of the indicators measuring a particular factor (Fornell and Larcker, 1981). Both the composite reliability and the variance extracted estimates are shown in Table 1. Fornell and Larcker (1981) recommend a minimum composite reliability of .60. An examination of the composite reliabilities revealed that all meet that minimum acceptable level.
The variance extracted estimates assesses the amount of variance that is explained by an underlying factor in relation to the amount of variance due to measurement error. For instance, the variance estimate for F1 was 0.838, meaning that 83.8% of the variance is explained by the F1 construct, and 16.2% is due to measurement error. Fornell and Larcker (1981) suggest that constructs should exhibit estimates of .50 or larger. Estimates less than .50 indicate that variance due to measurement error is larger than the variance captured by the factor. The variance extracted estimates all meet this minimum threshold, so the validity of the latent construct as well as the associated constructs is acceptable. It should also be noted that Hatcher (1994), cautions that the variances extracted estimate test is conservative; reliabilities can be acceptable even if variances extracted estimates are less than .50.
Convergent validity is present when different instruments are used to measure the same construct and scores from these different instruments are strongly correlated. In contrast, discriminant validity is present when different instruments are used to measure different constructs and the measures of these different constructs are weakly correlated.
In the present study, convergent validity was assessed by reviewing the t-tests for the factor loadings. If all the factor loadings for the indicators were greater than twice their standard errors, the parameter estimates demonstrated convergent validity. That all t-tests are significant showed that all indicators were effectively measuring the same construct (Anderson & Gerbing, 1988). Consider the convergent validity of the ten indicators that measure F1. The results show that the t-values for these ten indicators range from -14.480 to 18.510. These results support the convergent validity of Sympathique, Desagreable, Amicale, Souple, Severe, Autoritaire, Compatissante, au coeur tender, Spontanée, Distante, and Attentive aux autres as measures of F1.
Discriminant validity was assessed through the use of variance extracted test. Constructs were evaluated by comparing the variance extracted estimates for two factors, and then compared with the square of the correlation between the two factors. Discriminant validity is demonstrated if both variance extracted estimates are greater than the squared correlation. In the present study, the correlation between the factors F1 and F2 was 0.154; the squared correlation was 0.024. The correlations and squared correlations are shown in Table 2. The variance extracted estimate was 0.838 for F1 and 0.666 for F2. Because the variance extracted estimates are greater than the square of the interfactor correlation, the test supports the discriminant validity of these two factors. Examination of the other variance extracted estimates and squared correlation coefficients supported discriminant validity within the model.
References
Anderson, J.C. & Gerbing, D.W. (1988). Structural equation modeling in practice: A
review and recommended two-step approach. Psychological Bulletin, 103, 411-423.
Bollen, K.A. (1989). Structural equations with latent variables. New York: John Wiley
& Sons.
Fornell, C. & Larcker, D.F. (1981). Evaluating structural equation models with
unobservable variables and measurement error. Journal of Marketing Research, 18,
39-50.
Hatcher, L. (1994) A step-by-step approach to using SAS for factor analysis and
structural equation modeling. Cary, NC: SAS Institute Inc.
Jöreskog, K.G. & Sörbom, D. (1989). LISREL 7: A guide to the program and
application, 2nd edition. Chicago: SPSS Inc.
Dissertation Statistics Samples
Everybody knows how tiring it is to write a dissertation. Completing a dissertation is the final and most important task in receiving your degree. A dissertation is a lengthy piece of discourse imparting new views, approaches and findings on any topic from any field. One does not really feel good about facing hurdles and problems while going for their degree, and writing a dissertation is the biggest hurdle one can ever face. The task of authoring a dissertation is very complex, and it demands dedication and hard work on the part of students. Students learn and study the entire year, yet it is often in the writing of a dissertation that they have many, many unanswered questions. It is very unfortunate that at such crucial times, faculty and advisors are unavailable and help is very scarce. It is in these times— when faculty and advisors are not available, when unanswered questions about dissertations continually discourage students, when the pressure continues to mount on students as the work in writing their dissertations becomes more and more tedious and time consuming— that statistics samples can rescue struggling students in need of help and guidance.
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While students get step-by-step guidance from dissertation statistics samples, they also get constructive ideas as to how to enhance the uniqueness of their own dissertation. Consulting dissertation statistics samples does not mean that students are copying ideas and theory from existing dissertation statistics samples. Rather, it means that by having a look at dissertation statistics samples, students get ideas as to how to make their own dissertation an error free dissertation. Dissertation Statistics samples basically help the students in doing the statistics part of their dissertation. With the help of dissertation statistics samples, students can inquire about and interpret their statistical issues with an approved form of guidance. Through dissertation statistics samples, students can approach their own dissertations with a new and unique perspective. It helps them to edit and proofread their own dissertation, and finally, it helps them to feel confident that their own dissertation will be accepted and approved.
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A dissertation statistic sample is a tool that can help as it is much like an example of another dissertation. They are copies of approved and successful dissertations statistics samples. A dissertation statistics sample is the help that every dissertation-writing student requires to overcome his/her stress of attaining a degree. Dissertation statistics samples can be acquired from various services that cater to this particular need of dissertation statistics sample. Getting your dissertation is ensured when you have some dissertation statistics samples in your grasp. Additionally, dissertation statistics samples are very helpful for those “first timers” who have never written a dissertation before.
Though students are well read and perform research the entire year, they can struggle in deciding upon a particular topic on which to base their dissertation. Dissertation statistics samples provide these students with many examples and the samples can help them come up with an idea for a topic. Then, the dissertation statistics samples give them ideas about the dissertation proposals, which involve extensive research on the part of the students if they do not have any guidance from dissertation statistics samples. Dissertation statistics samples are dissertations that have been approved, signed and published by proper authorities, and they are always helpful to students. They are helpful because after reviewing the sample dissertation statistics samples, students will be better able to write their own dissertation and check for faults in their own ideas.
While students get step-by-step guidance from dissertation statistics samples, they also get constructive ideas as to how to enhance the uniqueness of their own dissertation. Consulting dissertation statistics samples does not mean that students are copying ideas and theory from existing dissertation statistics samples. Rather, it means that by having a look at dissertation statistics samples, students get ideas as to how to make their own dissertation an error free dissertation. Dissertation Statistics samples basically help the students in doing the statistics part of their dissertation. With the help of dissertation statistics samples, students can inquire about and interpret their statistical issues with an approved form of guidance. Through dissertation statistics samples, students can approach their own dissertations with a new and unique perspective. It helps them to edit and proofread their own dissertation, and finally, it helps them to feel confident that their own dissertation will be accepted and approved.
Consulting dissertation statistics samples is becoming more and more popular. Students must perform at top level and to do so, they must complete a top-notch dissertation. In order to finish in a timely manner, they must seek professional help and guidance. Using dissertation statistics samples is one such means of getting help. With this guidance, students will no-doubt achieve what they sought out to achieve when they began their studies.
Thursday, March 26, 2009
Run Test of Randomness
Run test of randomness is a non parametric test that is widely used to test the randomness of a sample. While run test of randomness is a sufficient test, it does not necessarily give an exact result in all cases. For example, this can be used in the stock market. If we want to test whether prices of a particular company are behaving randomly or if there are any patterns in the price of that company, we can use the run test of randomness. Or, in another case, if we want to test whether or not a sample is independent of each other or if the sample has any pattern, we can use the run test of randomness. Thus, the test for such problems is called the Run test of randomness.
Key concept and terms:
Run: Run is basically the newly assigned value to a part of a particular series. For instance, if in a sample M= male and F=female, the first 22 responses in that sample might come as MMMMFFFMFFFFMFFFFMMMFF. Starting from MMMM and ending with FF, there are 8 runs in this example. Basically, run test for randomness assumes binary value for that particular series. Run test for randomness assumes that the value of the binary variable must be equal to 2 binary values or more than 2 values. In SPSS, run test of randomness can test many values in a single time but that value must be numeric or we should convert them into numeric form.
Run Test: Run test is based on the law of probability. Run test of randomness can be performed in SPSS very easily. SPSS computes observer run and gives a critical value for run. We can compare that observed value with the computed critical value. SPSS shows two tailed test value by default. For a small sample, binary variable exact test is available to test its significance. For the larger sample, Monte Carlo estimation gives the significant value to test its randomness.
Cut Point: The algorithm of Run test for randomness divides the series through cut points. We can select cut point mean, median, or mode to specify as a custom point.
Type of significance estimate: Run test of randomness significance can be easily tested by using an exact button in SPSS available in run test.
Assumptions in Run test of randomness:
1. Data order: run test of randomness assumes that data is entered in order (not grouped).
2. Numeric data: Run test of randomness assumes that data is in numeric form. This is a compulsory condition for run test, because in numeric form, it is easy to assign run to that particular value.
3. Data Level: In run test or randomness we assume that data should be in order. But if data is not in ordered form, then the researcher has to assign a value. These values are one of the following: mean, median, mode or a cut point. By assigning one of these values, data can be ordered.
4. Distribution: Run test of randomness is a non-parametric test. Hence this test does not assume any distribution like any other parametric test.
Run Test in SPSS: Run test in SPSS is available in a non-parametric test in the analysis menu. By selecting this option, we will drag the variable in to the test variable list and select the “cut point” option. After clicking the “ok” button, the result will come in front of us. By examining the significance value, we can accept or reject the null hypothesis.
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Wednesday, March 25, 2009
Dissertation Statistics Services
Writing a dissertation is a very significant part of a student’s academic life. Writing an error and stress-free dissertation is what every student desires. This, however, is very difficult, because it’s almost impossible for an inexperienced and busy student to overcome the doubts that always surround a dissertation. In fact, the very word “dissertation” is enough to strike fear in most students, as the process of writing a dissertation is that stressful. At such stressful times, students would benefit greatly from having a helping hand that can take care of the queries surrounding the writing of the dissertation. Without help, the unsuccessful search for solutions of queries gets intensified because students do not have access to their advisors when they need it. To avoid such confusion and stress, it is advisable for students to get help from dissertation statistics services.
Dissertation statistics services provide professionals who can help you step-by-step as you have questions about writing the dissertation. Professionals from dissertation statistics services get the correct methodology for your dissertation. Along with the help in the collection of data, dissertation statistics services workers help you decide the most appropriate statistical test required to analyze your data. The choice of the statistical test is determined by the salient features of your study and research design. Dissertation statistics services and consultants are well aware of the importance of dissertation help. The consultants from dissertation statistics services make sure that you have the basic understanding of your study and data and know how to comprehend this data. This is very important as it will be used to decide the most appropriate statistical test to answer your dissertation and research question. Just like computers, statistical tests are mathematical formulas that can easily be implemented with the help provided by the dissertation statistics services. Finally, the consultant from dissertation statistics services can help you in determining whether the used statistical tests are appropriate to the design of your dissertation or research study.
Dissertation statistics services help you in many ways:
Ø They give guidelines in developing a dissertation from the beginning.
Ø They write the method section of your dissertation.
Ø They design, analyze, calculate and write the statistical portion of your method section.
Ø They draft and develop a plan for data analysis.
Ø They perform a power analysis for the required sample size.
Ø They analyze and transcribe your data.
Ø They help you write your result section.
Ø They attend to the comments given by the advisors and reviewing committee.
Ø They help you build up a defense for the regular dissertation comments.
Thus, dissertation statistics services can explain the purpose and logic of your dissertation. You can discuss your choice of your dissertation measures and instruments very openly with dissertation statistics services. You can also do the rehearsing for the reasons for your choice of specific statistics for your dissertation. With the consultant sent by the dissertation statistics services, you can practice the discussion of your dissertation findings.
Hiring the help of dissertation statistics services is very popular as they offer the most sought after methods for students who are going through the difficult processes of dissertation statistics. While acquiring the dissertation statistics services, students should be aware of their market value and should make sure that the dissertation statistics services are credible, available and that the dissertation statistics services are well established. In this world where one often struggles to differentiate between what is fake and what is real, it is a very important task to decide on the proper dissertation statistics service provider. Despite the difficulty one might have in choosing the proper dissertation statistics service, the service is indeed very important and necessary to the process of writing a dissertation. Further, dissertation statistics services can be very cost effective, considering how much time is saved by having the dissertation statistics service. Most services are very economical and well worth it. Finally, most statistics dissertation services provide the first consultation session free of charge, and this should also help in choosing the proper service to fit the student’s or organization’s needs.
Thursday, March 19, 2009
ANCOVA Assumptions
While examining the differences in the mean values of the dependent variable related to the effect of controlled independent variables, it becomes necessary to take into account the influence of uncontrolled independent variables. In such cases, Analysis of Covariance (ANCOVA) assumptions are used. In ANCOVA, assumptions include at least one categorical independent variable and at least one interval or metric independent variable. The categorical independent variable is called a factor, whereas the metric independent variable is called a covariate.
In ANCOVA assumptions, the most common use of covariate is to remove extraneous variations from the dependent variable. This is because in ANCOVA assumptions, the effect of factors is of major concern.
Like ANOVA, ANCOVA assumptions have similar assumptions. These assumptions are as follows:
The variance that is being analyzed or estimated should be independent, which also holds true for ANCOVA assumptions.
In ANOVA, the variable which is dependent in nature must have the same variance in each category of the independent variable. In the case of more than one independent variable, the variance must be homogeneous in nature, within each cells formed by the independent categorical variables, which also holds true for ANCOVA assumptions.
In ANOVA, it is assumed that the data upon which the significance test is conducted is obtained by random sampling, which also holds true for ANCOVA assumptions.
When analysis of variance is conducted on two or more factors, interactions can arise. An interaction occurs when the effect of independent variables on a dependent variable is different for different categories, or levels of another independent variable. If the interaction is significant, then the interaction may be ordinal or disordinal. Disordinal interaction may be of a no crossover or crossover type. In the case of the balanced designs, while conducting ANCOVA assumptions, the relative importance of factors in explaining the variation in the dependent variable is measured by omega squared. Multiple comparisons in the form of a priori or a posteriori contrast can be used for examining differences among specific means in ANCOVA assumptions.
ANCOVA assumptions also assume some other assumptions apart from the assumptions made in ANOVA.
In ANCOVA assumptions, the adjusted treatment means the computed or the estimated are based on the fact that the variable by covariate interaction is negligible. If this ANCOVA assumption is violated, then the adjustment of the response variable to a common value of the covariate will be misleading.
ANCOVA assumptions combine with the assumption of linear regression. The method of ANCOVA assumptions is done by using a linear regression. So in ANCOVA assumptions, the relationship between the independent and dependent variable must be linear in the parameters. Thus, in ANCOVA assumptions, the different levels of the independent variable will follow normal distribution with mean zero.
ANCOVA assumptions also assume the homogeneity of regression coefficients which is based on the fact that the regression coefficient for every group present in the data of the independent variable should be same. If this fact of ANCOVA assumptions is violated, then the ANCOVA assumption will be misleading.
Wednesday, March 18, 2009
Validity
Validity means accurate or error free conclusion(s) from the data. Technically, we can say that a measure leads to valid conclusion from a sample that can be taken as valid inference about the population. When we talk about validity, we are talking about four major types:
1. Internal validity
2. External validity
3. Statistical conclusion validity
4. Construct validity
Internal validity: When the relationship between the variable is casual, than it is called internal validity. Internal validity refers to the casual relationship between the dependent and the independent variable. In internal validity, we are concerned with the factor responsible for change in the dependent variable. It is related to the design of the experiment, such as when it is used for the random assignment of treatments.
External validity: External validity is when there is a casual relationship between the cause and effect that can be generalized or transferred to different people, to different treatment variables and to different measurement variables.
Statistical conclusion validity: Statistical conclusion validity occurs when we talk about the inference about the degree of the relationship between the two variables. For example, it is used when two variables are studied and we want to draw a conclusion about the strength of the relationship between the variables. When we arrive at the correct decision about the strength of the relationship for both of the variables, then it is said to be statistical conclusion validity. Statistical conclusion validity has two major types of errors:
Type one error: Type one error occurs when we accept the hypothesis, but that hypothesis is inaccurate. It also occurs when we say that there is a relationship between the two variables but in reality there is no relationship between them.
Type two error: Type two error occurs when we reject the hypothesis that is true or when there is no relationship between variables, yet we say that the relationship exists.
Power analysis is used to detect the relationship in statistical conclusion validity. When we are using statistical conclusion validity, we come across several problems. One of these problems occurs when we use a small sample size. In a small sample size, there is a possibility that the result will not be accurate. To overcome this problem, the sample size should be increased. Violation of the statistical assumption is also a threat for statistical validity. If we use a biased value in analysis, then the results may not be accurate. If the wrong statistical test is applied, then the conclusion may not be accurate.
Construct validity: Construct validity is when the construct is involved in predicating the relationship for the dependent variable. For example, in structural equation modeling, when we draw the construct, then we assume that the factor loading for the construct should be greater than .7. Cronbach's alpha is used to draw the construct validity. .60 is considered acceptable for exploratory purposes, .70 is considered adequate for confirmatory purposes, and .80 is considered good for confirmatory purposes. If the construct satisfies the above assumption, then the construct will contribute in predicting the relationship for dependent variables. Convergent/divergent validation and factor analysis is also used to test construct validity.
Relationship between reliability and validity: A test that is unreliable cannot be valid and a test that is valid must be reliable. Reliability is necessary but not a sufficient condition for validity. Thus, validity plays a major role in analysis and in making accurate decisions.
The following are overall validity threats:
1. Insufficient data collected to make valid conclusions
2. Measurement done with too few measurement variables
3. Too much variation in data or outlier in data
4. Wrong selection of sample
5. Inaccurate measurement method taken for analysis
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Thursday, March 12, 2009
Mann-Whitney in SPSS
Mann-Whitney in SPSS is the most widely used non-parametric test, and it is used as an alternative to the t-test. In the Mann-Whitney in SPSS, we do not make assumptions about the parent population as in the t-test. Mann-Whitney in SPSS tests that the two sample populations are equivalent in location. The observations from both the groups are combined together and are ranked. In the case of ties in the Mann-Whitney in SPSS, the average rank is obtained. In the Mann-Whitney in SPSS, one should keep the number of ties relatively small in relation to the total number of observations.
If the populations are identical in location, the ranks should be randomly mixed between the two samples. The test calculates the number of times a score from group 1 precedes a score from group 2, and the number of times a score in group 2 precedes a score from group 1. The value of the Mann-Whitney in SPSS is the one that comes out to be the smaller of these two numbers.
While conducting the Mann-Whitney in SPSS, one needs to perform the following options: go to “Analyze menu” and click on the “non parametric tests” option, select the “Two independent sample tests” option, and select the test type (Mann-Whitney in this case).
The following are the operations which SPSS does while calculating the Mann-Whitney in SPSS:
· We rank the cases in order of increasing size, and the test statistic U, which indicates the number of times that a score from group 1 preceded a score from group 2.
· We can compute an exact level of significance if there are fewer cases. When there are more than just a few cases, we transform U into Z statistic, and a normal approximation p value is computed.
· A test statistic is then calculated for each variable.
In order to compute the Mann-Whitney in SPSS, the following actions need to be performed:
Let xi (i=1…n1) and yj (j=1…n2) be an independent sample of n1 and n2 from the population probability density f ( ) and f2 ( ) respectively. If we want to test the null hypothesis H1 : f1 ( ) = f2 ( ), let T be the sum of ranks of the y’s in the combined ordered sample. The test statistic U is defined in terms of T as follows:
U= n1 n2+ n2 (n2 + 1)/2 – T
If T is significantly larger or smaller, then the null hypothesis is rejected. The problem is finding the distribution of T under null hypothesis. Unfortunately, it is very troublesome to obtain the distribution of T under null hypothesis. However, Mann-Whitney in SPSS has obtained the distribution of T for small n1 and n2 and has shown that T is asymptotically normal. It has been established that under null hypothesis, U is asymptotically normally distributed as N (µ, σ2), where
µ=E (U) = n1 n2/2 and σ2= V (U) = n1 n2(n1 + n2+1)/12.
Asymptotic normal means that the true parameter approaches the normal distribution as the size of the sample increases.
Here, Z= U-µ/ σ which is asymptotically normal with mean 0 and variance 1.The approximation of Mann-Whitney in SPSS is fairly good if both n1 and n2 are greater than 8. This means that the size of the two independent samples should be greater than 8, and then only the approximation given by Mann-Whitney in SPSS is true.
The Asymptotic Relative Efficiency (ARE) of the Mann-Whitney in SPSS is relative to the two sample t-tests, which is greater than or equal to 0.864. For a normal population, the ARE is 0.955. Accordingly, the Mann-Whitney in SPSS is regarded as the best non parametric test for location. Asymptotic Relative Efficiency (ARE) means that it is the limit of the relative efficiency, as the size of the sample increases.
Independent and Dependent Variables
In order to know independent and dependent variables, one should know what variables are. Variables are properties or characteristics of some event, object, or person that can take on different values or amounts. When researchers are conducting research, they often manipulate variables.
Now, let us discuss independent and dependent variables in detail:
Independent variable(s) are the variables or alternatives that are manipulated (i.e. the level of these variables are changed by a researcher) and whose effects are measured and compared. They are also called Predictor(s), as they predict the values of the dependent variable or predicted variables in the model. In layman’s language, the independent variable is a variable that stands alone and is not changed by the other variable one is trying to measure. For example, while looking at someone’s age, variables like what a person eats, how much he watches television etc... do not change the person’s age. That is why they are called the other variables. In fact, when one is looking for some kind of a relationship between the variables, then one is trying to see if the independent variable causes some kind of change in the other variables.
The other variable(s) can also be called dependent variable(s). As the name suggests, they are the variables that measure the effect of the independent variable(s) on the test units. In layman’s language, the dependent variables are the variables which are completely dependent on the independent variable(s). They are also called Predicted variable(s) as they are the values to be predicted or assumed by the predictor / independent variables. For example, a student’s score could be a dependent variable because it could change depending on several factors such as how much he studied, how much sleep he got the night before he took the test, or even how hungry he was when he took the test. Usually, when one is looking for a relationship between two things, one is trying to find out what makes the dependent variable change the way it does.
Independent variables are also called “regressors,” “controlled variable,” “manipulated variable,” “explanatory variable,” “exposure variable,” and/or “input variable.” Similarly, dependent variables are also called "response variable," "regressand," "measured variable," "observed variable," "responding variable," "explained variable," "outcome variable," "experimental variable," and/or "output variable."
A few examples can highlight the importance and usage of dependent and independent variables in a broader sense:
If one wants to measure the influence of different quantities of nutrient intake on the growth of an infant, then the amount of nutrient intake can be the independent variable, while the dependent variable can be the growth of an infant measured by height, weight or other factor(s) as per the requirement of the experiment.
If one wants to estimate the cost of living for an individual, then factors such as salary, age, marital status etc. are independent variables. The cost of living for a person is highly dependent on such factors, hence can be designated as the dependent variable.
In the case of the time series analysis, forecasting a price value of a particular commodity is again dependent on various factors as per the study. Suppose we want to forecast the value of gold, for example. In such an instance, the seasonal factor can be an independent variable on which the price value of gold will depend.
In the case of a poor performance of a student in an examination, the independent variables can be the factors like the student not attending classes regularly, the student having poor memory etc., which can reflect the grade of the student. Here, the dependent variable is the test score of the student.
Tuesday, March 10, 2009
Sample Dissertation Statistics
A sample of dissertation statistics is a template that dissertation and thesis students can use to help present their findings. This template can be invaluable to students when it comes to working on, writing and finishing a dissertation. This can be especially useful as writing a dissertation is not an easy task and it requires hard work combined with efficiency on the part of the student. And while the most important thing for a graduate or doctoral candidate is the completion of his or her dissertation, the dissertation comes at the end of an academic period, a time when everything seems important yet impossible to finish. Though they may have studied and researched throughout the year, students face problems making sense of their own ideas, even when it comes to deciding the very topic of their dissertation. One aspect of a quantitative dissertation is the Dissertation Statistics. Most students are “first timers” and need thorough and expert guidance from someone skilled. What’s more, teachers may not be available to the students all the time, and in fact, may have even less time at the end of the semester. Dissertation statistics samples can be a blessing for students because most students are not familiar with the writing and formatting skills of thesis research and writing. Dissertation statistics samples can be used by the students as a reference.
Dissertation Statistics Samples can be used for writing the dissertation proposal as well as the actual dissertation itself. Not only can dissertation statistics samples help by assisting the student in deciding a topic, dissertation statistics samples can also be useful when it comes to the terminology and writing style that should be used for the student’s dissertation. Looking at dissertation statistics samples is practical as the samples provide an idea of the research and writing methodology as well as examples of the construction of other parts of the entire dissertation. This can be extremely valuable as it can help in increasing the overall quality and reliability of the student’s own dissertation.
In considering the benefits of consulting dissertation statistics samples, a student may wonder if it is worth it and if his or her dissertation will truly benefit from the use of the dissertation statistics samples. However, this student must remember that the task of writing a dissertation can become much easier with the use of samples, which can provide useful insight in terms of common pitfalls to avoid. Additionally, a student writing a dissertation is likely to have many concerns about things such as graphs, tables, calculations, literature review, citation formats, etc… and the dissertation statistics samples can easily provide solutions to such problems, as long as they come from a reliable source.
Dissertation statistics samples can provide help in many different ways. To begin with, they provide a benchmark on which the student’s entire dissertation can be based. This makes the entire research project much less intimidating. Dissertation statistics samples can also be used as templates to help edit and “clean-up” an already prepared piece of text. It can do the same for tables, charts and graphs. In regards to the interpretation of all aspects of the dissertation, dissertation statistics samples consulting can advise and guide the student in terms of theme, style and format. On another note, they can also lend suggestions as to which statistical procedures are most useful to a specific kind of dissertation. Most importantly, dissertation statistics samples can help the students find their errors and they can suggest improvements. Finally, because dissertation statistics samples come from a reliable source and are in principle, pre-approved, students get the benefit of having a good standard to live up to.
In consulting dissertation statistics samples, students should remember that in terms of guidance, it is most useful to consult more than one dissertation statistics sample. However, using too many dissertation statistics samples can also lead to information overload. It is best to browse through a few dissertation statistics samples and select the right areas and sections for reference. Also, it is important to remember that though dissertation statistics samples can provide the much-needed guidelines, they cannot provide the actual content.
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