The underlying factors of depression

Transcription

1 The underlying factors of depression A factor analysis on the items of the IDS-C 30 Bachelor thesis in Mathematics January 2015 Student: Leslie Zwerwer, s Primary supervisor: prof. dr. E.C. Wit Secondary supervisor: dr. W.P. Krijnen

2 THE UNDERLYING FACTORS OF DEPRESSION 2 Abstract In this research we investigated the underlying factors of depression. The sample contained 2981 subjects with the following characteristics: "healthy", suffering from Major Depressive Disorder or suffering from Bipolar II Disorder. They completed the Inventory of Depressive Symptomatology-Clinican 30 (IDS-C 30 ), three times in a half year period. During this period the subjects received Cognitive Behavioural Therapy. To find the underlying factors we used different methods. First we generated three models using exploratory factor analysis. Subsequently we tested these models using confirmatory factor analysis. We also tested a model that was suggested by the IDS-C 30. We found that the model containing the factors anxiety, gloominess and somatic/vegetative was on average the best predicting and the most correct model.

3 THE UNDERLYING FACTORS OF DEPRESSION 3 Contents Introduction 4 Methods 5 Inventory of Depressive Symptomatology-Clinican Initial data analysis Exploratory factor analysis Interpretation Estimating scores on factors or variables Confirmatory factor analysis Goodness of fit indices Confirmatory factor analysis with ordinal data Measurement invariance Missing values Results 28 Results exploratory factor analysis Results confirmatory factor analysis Results confirmatory factor analysis with ordered variables Results measurement invariance Results multiple imputation Discussion 35 Discussion exploratory factor analysis Discussion confirmatory factor analysis Discussion measurement invariance Discussion multiple imputation Some suggestions for further research References 41 Appendices 43 Appendix A Appendix B Appendix C

4 THE UNDERLYING FACTORS OF DEPRESSION 4 Introduction Almost everyone has dealt with depression in some way. It is the most common psychological illness in the world. According to the World Health Organization ( globally more than 350 million people are suffering from depression. Moreover it is the most common cause of disability in the world and a major contributor to the global burden of diseases. Depression is a rich concept and there are a lot of unanswered questions about this psychological illness. For example: why are some people more sensitive to develop a depression than others? And why does the length of a depression differ among people? Therefore it is important to research the different symptoms and causes of depression. There are different kinds of depression. The three most important kinds are: Major Depressive Disorder (MDD), Dysthemic Disorder (DD) and Bipolar Disorder (BD). In this article we focus on the Major Depressive Disorder and Bipolar II Disorder. According to the DSM-IV (American Psychiatric Association, 1994, pp.327), "someone is suffering from Major Depressive Disorder if he is experiencing five (or more) of the following symptoms. These symptoms have to be present during the same 2-week period and they represent a change from previous functioning. Moreover at least one of the symptoms is either (1) depressed mood or (2) loss of interest or pleasure. Note: symptoms that are clearly due to a general medical condition or mood-incongruent delusions or hallucinations are excluded. 1. depressed mood most of the day, nearly every day. 2. markedly diminished interest or pleasure in all, or almost all, activities most of the day, nearly every day. 3. significant weight loss when not dieting or weight gain (e.g., a change of more than 5% of body weight in a month), or decrease or increase in appetite nearly every day. 4. insomnia or hypersomnia nearly every day. 5. psychomotor agitation or retardation nearly every day. 6. fatigue or loss of energy nearly every day. 7. feelings of worthlessness or excessive or inappropriate guilt (which may be delusional) nearly every day. 8. diminished ability to think or concentrate, or indecisiveness, nearly every day. 9. recurrent thoughts of death (not just fear of dying), recurrent suicidal ideation without a specific plan, or a suicide attempt or a specific plan for committing suicide. Moreover the symptoms should not meet criteria for a Mixed Episode. The symptoms cause clinically significant distress or impairment in social, occupational, or other important areas of functioning. Furthermore the symptoms are not due to the direct physiological effects of a substance (e.g., a drug of abuse, a medication) or a general medical condition (e.g., hypothyroidism). Last, the symptoms are not better accounted for by bereavement, i.e., after the loss of a loved one, the symptoms persist for longer

5 THE UNDERLYING FACTORS OF DEPRESSION 5 than 2 months or are characterized by marked functional impairment, morbid preoccupation with worthlessness, suicidal ideation, psychotic symptoms, or psychomotor retardation." Bipolar II Disorder differs in some ways from Major Depressive Disorder. According to the DSM-IV (American Psychiatric Association, 1994, pp. 362), someone is suffering from Bipolar II Disorder if he is experiencing all of the following symptoms: 1. Presence (or history) of one or more Major Depressive Episodes. 2. Presence (or history) of at least one Hypomanic Episode. 3. There has never been a Manic Episode or a Mixed Episode. 4. The mood symptoms in Criteria (1) and (2) are not better accounted for by Schizoaffective Disorder and are not superimposed on Schizophrenia, Schizophreniform Disorder, Delusional Disorder, or Psychotic Disorder Not Otherwise Specified. 5. The symptoms cause clinically significant distress or impairment in social, occupational, or other important areas of functioning. A Hypomanic Episode is defined as a distinct period during which there is an abnormally and persistently elevated, expansive, or irritable mood that lasts at least four days. Moreover a manic episode is defined by a distinct period during which there is an abnormally and persistently elevated, expansive, or irritable mood. This period of abnormal mood must last at least one week. Furthermore a Mixed Episode is characterized by a period of time (lasting at least 1 week) in which the criteria are met both for a Manic Episode and for a Major Depressive Episode nearly every day. In this article we will start discussing the initial data analysis. Moreover we will explain exploratory factor analysis and confirmatory factor analysis for continuous variables. We will continue with a subsection about goodness of fit indices and a subsection about confirmatory factor analysis for ordinal variables. Subsequently we will explain the concepts of measurement invariance and multiple imputation. After that the results of the different analyses are shown. In the last section we will discuss these results. Methods Inventory of Depressive Symptomatology-Clinican 30 In this research we used data from Nesda ("Nederlandse Study naar Depressie en Angst", translates as "The Dutch Study of Depression and Anxiety") (Boschloo et al., 2014). The Inventory of Depressive Symptomatology-Clinican 30 (IDS-C 30 ) was filled in by 2981 subjects. There were three kinds of subjects participating in this research: subjects suffering from major depression, subjects suffering from bipolar II disorder and subjects who did not suffer from depression. It would be interesting to look at the differences in the factor structure between these three groups, however this falls outside the scope of the thesis. At three different moments subject were asked to fill in the

6 THE UNDERLYING FACTORS OF DEPRESSION 6 IDS-C 30. The first moment was at the starting point of the cognitive behavioural therapy, the second and the third moment were respectively three months and six months later. The IDS-C 30 contains 30 items about criterion symptoms of depression, commonly associated symptoms and items relevant to melancholic or atypical symptom features. All DSM-IV criterion items for major depressive episodes are included in the test. There is evidence of the psychometric properties of the IDS-C 30 for depressed inpatients (Corruble, Legrand, Duret, Charles & Guelfi, 1999) as well as depressed outpatients (Trivedi et. al., 2004). Moreover it can be used to evaluate depressive symptom severity (Rush, Gullion, Basco, Jarrett & Trivedi, 1996). Furthermore factor analysis on the IDS, Hamilton Depression Rating Scale (HDRS) and Beck Depression Inventory (BDI) has revealed that the IDS provided more complete factor coverage than the HDRS or BDI did (Gullion & Rush, 1998). There are four options for every item in the IDS-C 30. Each item can be rated from 0 to 3, where 0 is indicating no suffering and 3 is indicating extreme suffering from this symptom. The IDS-C 30 was not completed by all subjects at all three measurements. Some subjects participated just in one or two measurements. One can find the IDS-C 30 in Appendix B. To complete the IDS-C 30 one needs to answer 28 items, because items 11 and 12, and items 13 and 14 are each other opposites. Therefore one needs to complete either item 11 or 12 and either item 13 or 14. We will refer to items 11 or 12 as item 11 and to items 13 or 14 as item 12. Moreover we will refer to items number as respectively items Items number 1 4 were concerned with sleep. Items number 5 10 were concerned with mood. Items number 12, and 28 were concerned with evaluating somatic symptoms. The symptoms concerning appetite were measured with item 11. The mental symptoms of depression were measured with items and item 27. Initial data analysis The frequencies of the answers to the different questions are shown in figure 1, figure 2 and figure 3. It is easy to see that the most common answer was 0. So the majority of the subjects did not experience any suffering from the symptoms. The most rare answer was option 3, indicating that most subjects did not experience severe suffering from the symptoms. We now look more precisely at figure 1. We will refer to the first, second and third measurement moment, using respectively T1, T2 and T3. Item number 9 was the most unanswered item. This could be due to comprehensibility of the question or the item was too confronting for some subjects. Also items number 3, 8 and 20 were more often unanswered than other items. Item number 28 was filled in by every subject who participated at T1. Furthermore item number 14 shows a different pattern with respect to the other items. When we look at the frequency of answers in item 14, one can see that far more people answered option 3 than 2. At T1 there were 2947 participants and 2640 participants answered all items (i.e. there were 307 subjects with missing values). The frequencies of the answers to the different questions with respect to T2 are

7 THE UNDERLYING FACTORS OF DEPRESSION 7 shown in figure 2. In accordance with the first measurement item number 9 was the most unanswered item. Also items number 3, 8 and 20 were more often unanswered than other items. Items number 1 and 5 were most frequently filled in. Notice that item 14 has the same pattern as at T1. At T2 there were 2444 participants and 2108 subjects answered all items (i.e. there were 336 subjects with missing values). First measurement Items (1 28) Figure 1. : Frequencies of the answers at the first measurement moment (T1) Second measurement Items (1 28) Figure 2. : Frequencies of the answers at the second measurement moment (T2) The frequencies of the answers to the different questions with respect to T3 are shown in figure 3. In accordance with T1 and T2 item number 9 was the most unanswered item. Also item number 8 was more often unanswered than other items. Item

8 THE UNDERLYING FACTORS OF DEPRESSION 8 number 6 was completed by almost every subject participating at T3; just one participant did not answer this item. Notice that item 14 has the same pattern in T1, T2 and T3. In T3 there were 2505 participants and 2312 participants answered all items (i.e. there were 193 subjects with missing values). Third measurement Items (1 28) Figure 3. : Frequencies of the answers at the third measurement moment (T3) Now we added the answers of the items concerning sleep, mood, somatic, appetite and mental for every subject. We did this for all three measurement moments. We want to find correlations between these sum variables. We make use of Spearman s rank order, because all variables are ordinal. Spearman s rank order is a nonparametric technique. This means that it is not affected by the distribution of the population (Gauthier, 2001). The spearman rank order coefficient ranges from 1 to 1. The formula for spearman rank order is the following: r s = n n i=1 d2 i T x T y [ n ] [ T n 3 3 x 6 ] T y where n is the number of data pairs and d i is the difference between the rank of x i and y i. Moreover T x and T y are defined in the following way: Tx = Ty = n j=1 n j=1 t 3 j t j 12 t 3 j t j 12 (1) (2)

9 THE UNDERLYING FACTORS OF DEPRESSION 9 where t j is the numbers of ties in group j. Moreover equation 1 is for the x values and equation 2 is for the y values. Subsequently we calculated Spearman s rank order for the sum variables. We only used the complete cases. The results are presented in table 1, 2 and 3. Table 1: Spearman s rank order for the first measurement moment (T1) Appetite Mental Mood Sleep Somatic Appetite Mental Mood Sleep Somatic Table 2: Spearman s rank order for the second measurement moment (T2) Appetite Mental Mood Sleep Somatic Appetite Mental Mood Sleep Somatic Table 3: Spearman s rank order for the third measurement moment (T3) Appetite Mental Mood Sleep Somatic Appetite Mental Mood Sleep Somatic All sum variables have positive correlation between each other. We define a correlation of bigger than 0.70 as very large (cf. Hemphill, 2003). Therefore the following sum variables seem to be highly correlated ( r > 0.70) at T1: Somatic and Mental, Mood and Mental and Somatic and Mood. At T2 and T3 the correlations of Somatic and Mental and Mood and Mental" are very large. At all measurement moments the correlation of "Sleep and Appetite" seem to have a low correlation ( r < 0.30). One interesting thing is that all correlations of T2 and T3 are lower than the correlations of T1. It is possible that the dropouts had equality in terms of their rank score. If so the correlations will be lower at T2 and T3.

10 THE UNDERLYING FACTORS OF DEPRESSION 10 In figures 4, 5 and 6 one can see scatterplot matrices for all three measurement moments. The green lines are regression lines and the red lines are loess lines (i.e. a smooth line through the data). The positive diagonal contains a continuous line which represents the frequencies of answers to the sum variable. At T1 the global maximum is at every plot near the y-axis, so most subjects had a low score for all sum variables. The sum variables Appetite, Mental and Mood, have their global maximum directly at the lowest score possible and they slowly decrease. The sum variables Sleep and Somatic show a slightly different pattern. At first, both variables show an increasing pattern, when their maximum is reached they slowly decrease. So most subjects did report some suffering from symptoms concerning Sleep and Somatic. In figure 5 one can see a scatterplot matrix for T2. One can see that the positive diagonal of T2 looks a lot like the positive diagonal of T1. There are some slight changes however. At T1 after the global maximum the line decreased convex. At T2 the line decreases concave. So the decrease goes faster at T2. In figure 6 one can see the scatterplot of T3. The scatterplot matrix of T3 looks a lot like the scatterplot matrix of T2. appetite mental mood sleep somatic Figure 4. : Scatterplot matrix of the first measurement moment (T1) In table 4 one can see the percentage of missing values per item and per measurement. At T1 the percentage of missing values ranges from 1.141% to 2.583%, with a mean value of 1.680%. At T2 the percentage of missing values ranges from % to %, with a mean value of %. Finally the percentage of missing values for T3 ranges from % to %, with a mean value of %.

11 THE UNDERLYING FACTORS OF DEPRESSION 11 appetite mental mood sleep somatic Figure 5. : Scatterplot matrix of the second measurement moment (T2) appetite mental mood sleep somatic Figure 6. : Scatterplot matrix of the third measurement moment (T3)

12 THE UNDERLYING FACTORS OF DEPRESSION 12 Table 4: Percentage missing values Item Measurement moment 1 Measurement moment 2 Measurement moment 3 (T1) (T2) (T3) Exploratory factor analysis When the number of variables is large (e.g. 20 variables), there are a lot of correlations. Then it is quite difficult to summarize the representations of the pattern of correlations (Stevens, 1996). Therefore to discover the underlying processes of a construct one can use exploratory factor analysis. In factor analysis one searches in the set of variables for variables that form coherent subsets and are relatively independent of each other (Tabachnick & Fidell, 2014). Variables that have high correlation with one another, but have low correlations with other variables are combined into factors. Factors are linear combinations of the original variables. In this way we reduce the number of variables and due to these factors we can analyse the underlying processes of the construct. There are two kinds of factor analysis, namely exploratory factor analysis

13 THE UNDERLYING FACTORS OF DEPRESSION 13 and confirmatory factor analysis. In exploratory factor analysis the researcher tries to find out how many factors are present and whether these factors are correlated. Subsequently he will name the different factors. Therefore exploratory factor analysis is a hypothesis developing method. In confirmatory factor analysis the researcher tests his hypotheses about the number of factors that are involved and about the factor correlations. In confirmatory factor analysis the hypothesized model has a strong theoretical basis (Stevens, 1996). We will discuss confirmatory factor analysis more elaborate in section Results confirmatory factor analysis. In factor analysis one can see raw data of one subject in the following way: x i = τ + Λη η i + δ i where x i represents the data (a p 1 vector), τ is the intercept (a p 1 vector), Λ is called the factor loading matrix of dimension p m. Furthermore η is a m 1 vector which represents the scores on the factors and δ represents a p 1 vector with the person-specific scores on the factors. Moreover p is equal to the number of items and m equals the number of factors. To find the factor loading matrix, one must diagonalize the correlation or covariance matrix. For this we use the following theorem: Theorem. (Diagonalizability) Real symmetric matrices are diagonalizable by orthogonal matrices. Proof. We take a matrix A, an arbitrary real and symmetric matrix. We know that real symmetric matrices have real eigenvalues (proof in Appendix A). Consequently A is upper triangularizable by an orthogonal matrix P (Beezer, 2014). That means that there exists an orthogonal matrix P such that P T AP = U, where U is an upper triangular matrix. We now take the transpose of U. U T = P T A T P T T = P T AP = U So U T = U, this implies that U is a diagonal matrix. Therefore all real symmetric matrices are diagonalizable by orthogonal matrices. We use the notation S for the correlation or covariance matrix of the sample. The covariance matrix is a symmetric, matrix with non-negative eigenvalues. This means that this matrix is non-negative definite (Leon, 2010). Diagonalization of S will result in the following: L = V T SV where L is a matrix with the eigenvalues of S on the positive diagonal and zeros elsewhere. Furthermore the columns of V are equal to the eigenvectors of S. Moreover the first eigenvalue in L corresponds to the first eigenvector and so on. One can also rewrite this equation:

14 THE UNDERLYING FACTORS OF DEPRESSION 14 S = V LV T = V L 1 2 L 1 2 V T = ΛΛ T. So the matrix S is a product of two matrices. The matrix Λ is called the factor loading matrix. This matrix shows the correlations between the factors and the variables. The communality of a variable is equal to the sum of the squared correlation with each factor. This is equal to the variance in a variable accounted for by all the factors. It is always less or equal to 1. The goal of a factor analysis is to reduce the dimension, therefore it is useless to retain every eigenvalue of S (Thurstone, 1937). Kaiser (cf. Stevens, 1996) developed the following criterion: retain every eigenvalue that is greater than 1. The idea is the following: a factor explains as many items as the eigenvalue. If an eigenvalues is less than one, the factor explains less than one variable and therefore it does not summarize it. According to Stevens (1996) the Kaiser criterion is quite accurate when N > 250 and the mean communality is greater or equal than To make it more comprehensible, we will introduce an example (Tabachnick & Fidell, 2014). Lets assume, we have done a research about depression. We asked four questions concerning mood, mental state, somatic symptoms and sleep. The used data is artificial and not realistic. Subsequently we calculated the correlations S = If we diagonalize this, we get the following: S = As one can see the eigenvalues of the correlation matrix are 1.893, 1.447, and 0.185, because the last two eigenvalues are less than 1 we drop those eigenvalues and their associated eigenvectors. Therefore we get the following rank 2 decomposition: [ ] [ ] S = To find the factor loading matrix we need to take the square root of the

15 THE UNDERLYING FACTORS OF DEPRESSION 15 matrix of eigenvalues and multiply this matrix from the left with the matrix with the corresponding eigenvectors [ ] Λ = = So from the factor loading matrix we can deduce that the first factor and the variable mood have a correlation of Moreover the first factor and the variable mental have a correlation of Furthermore the first factor has a correlation of with the variable somatic and a correlation of with the variable sleep. The second factor and the variable mood have a correlation of The second factor has a correlation of with the variable mental and a correlation of with the variable somatic. Moreover the second factor has a correlation of with the variable sleep. To decide the number of factors to retain, one can use various methods. The Root Mean Square Error Approximation (RMSEA) is another example of a factor retention criterion (Preacher, Zhang, Kim, & Mels, 2013). The RMSEA is a goodness of fit criterion. It can be calculated in the following way: ( { FML RMSEA = max 1 }) df n, 0 where F ML is the minimized maximum likelihood function and df are the degrees of freedom of the chi-squared test. Moreover n is equal to the number of subjects. We will discuss the maximum likelihood function in section Results confirmatory factor analysis. The main goal of RMSEA is to maximize verisimilitude. It performs well for both small and big samples. The number of factors to be used according to the RMSEA are the smallest number of factors, where RMSEA drops below 0.05 (Preacher, Zhang, Kim, & Mels, 2013). Interpretation Suppose we have a factor loading matrix. To make the matrix easier to interpret, we can use rotation. The main goal of rotation is to maximize high correlation and minimize low correlations. This increases the interpretability of the factor loading matrix. One method to rotate is called varimax, which maximizes the variance of factor loadings. Varimax increases high factor loadings and decreases low ones. To accomplish this, the factor loading matrix Λ is multiplied from the right with a transformation matrix P. In the case of two factors matrix P is of the following form: [ ] cos θ sin θ P = sin θ cos θ

16 THE UNDERLYING FACTORS OF DEPRESSION 16 where θ corresponds to a rotation of θ of the factor axes about the origin. So in formulas we have the following: Λ rotated = ΛP. Now we proceed with our previous example: To improve the interpretability of the factor loading matrix we will use varimax rotation. We use a rotation of 65 degrees [ ] Λ rotated = cos(65 ) sin(65 ) = sin(65 ) cos(65 ) One can see that the first two variables have a high correlation with the second factor and the last two variables have high correlations with the first factor. The communality for the first variable is (0.034) 2 + (0.939) 2 = This means that 88.3% of the variance in mood is accounted for by factor 1 and factor 2. The Sum of Squared Loadings (SSL) are equal to the sum of the squares of every loading per factor. One can calculate the proportion of variance in the set of variables accounted for by the factors by dividing SSL by the number of variables. The proportion of covariance can be calculated by dividing the SSL of the factor by the sum of all SSL. So for our example we can now create the following table: Table 5: Factor loadings and communalities Factor 1 Factor 2 Communalities ( f 2 ) Mood Mental Somatic Sleep SSL Proportion of variance Proportion of covariance When varimax rotation is used, the factors will not correlate. An other rotation method is called promax. Promax rotation starts with an orthogonal rotated solution like varimax (Tabachnick & Fidel, 2014). These pattern/structure coefficients are then raised to some exponential power, k. When k is even, the signs of the resulting coefficients are restored after the exponential operation is performed (Thompson, 2004). These exponential operations result in lower factor loadings. Small factor loadings will tend to zero. This will result in a simple structure.

17 THE UNDERLYING FACTORS OF DEPRESSION 17 Estimating scores on factors or variables Suppose someone completed the test and we want to give the scores on the various factors. These scores can be computed by multiplying the standardized scores with a matrix C, called the factor coefficient matrix. The coefficient matrix C can be computed in the following way: C = S 1 Λ. We will use notation F for the matrix with the factor scores and Z for the matrix with the standardized scores on the test. To compute the factor scores one uses the following formula. F = ZC. We will now compute the factor scores of two subjects. We use the same artificial data as in our previous example. First we will calculate S 1 and then multiply it from the right by the matrix Λ S 1 = C = = So when we estimate the score on the first factor, the questions about mood, mental state, somatic and sleep are weighted respectively by: 0.081, 0.149, and For the second factor the questions about mood, mental state, somatic and sleep are weighted respectively 0.525, 0.542, and The standardized scores of the two subjects are shown below. Subsequently we calculate the factor scores. F = [ ] [ ] =

18 THE UNDERLYING FACTORS OF DEPRESSION 18 It is also possible to estimate someone s standardized scores on the test from the factor scores. One can compute the standardized scores in the following way: Z = F Λ T. Lets assume, we have the scores of four subjects on the factors and we want to estimate the scores on the test [ ] Z = = So if we multiply the scores on the factor by the right with the transpose of the factor loading matrix, we will get the estimations of the standardized scores. Confirmatory factor analysis Confirmatory factor analysis is a hypothesis testing procedure. The researcher will fix the number of factors and he will decide whether these factors are correlated or not (based on his hypotheses) (Brown, 2006). Moreover variables tend to load on just one factor. In confirmatory factor analysis we can split the variance in the following way (Mulaik, 2010): Total variance = True variance {}}{ common variance + specific variance + error variance }{{} Unique variance The true variance is equal to the common variance together with the specific variance. Each observed variable has a unique variance. The unique variance is equal to the specific variance together with the error variance. Moreover it is the part of the observed variable that is not explained by the factors (Stevens, 1996). Unique variance is sometimes called measurement error. Some assumptions for confirmatory factor analysis are: the model is specified in a proper way, the measurement errors are independent of the scores on the latent factors and the measurement errors are uncorrelated. The general goal of confirmatory factor analysis is to test the hypothesis that the observed covariance matrix for a set of measured variables is equal to the covariance matrix implied by an hypothesized model (Flora & Curan, 2004). So in confirmatory factor analysis the researcher will test the following: Σ = Σ(θ) where Σ is the population covariance matrix of the observed variables and Σ(θ) represents the population covariance matrix implied by θ, a vector with the model parameters. The covariance matrix implied by θ is equal to the expected value of xx T, where x = Λη + δ, cov(η, δ) = 0 and E(δ) = 0 (Bollen, 1989).

19 THE UNDERLYING FACTORS OF DEPRESSION 19 For simplicity we do not consider the intercept (τ ) in x. xx T = (Λη + δ)(λη + δ) T = (Λη + δ)(η T Λ T + δ T ) = Ληη T Λ T + Ληδ T + δη T Λ T + δδ T E(xx T ) = E(Ληη T Λ T + Ληδ T + δη T Λ T + δδ T ) = E(Ληη T Λ T ) + E(Ληδ T ) + E(δη T Λ T ) + E(δδ T ) = ΛE(ηη T )Λ T + ΛE(ηδ T ) + E(δη T )Λ T + Φ δ = ΛΦΛ T + ΛE(η)E(δ T ) + E(δ)E(η T )Λ T + Φ δ Σ(θ) = ΛΦΛ T + Φ δ where Φ contains the correlations among the factors and Φ δ is a diagonal matrix containing the error variances on the diagonal. In confirmatory factor analysis the researcher first specifies how the matrix Λ should look: which items should have loadings unequal to zero and which items should have loadings equal to zero (Stevens, 1996). Moreover he or she must specify if the factors should be correlated and these correlations or covariances should be estimated. Finally the measurement errors should be estimated. These estimations should be made based on the theory of the researcher. These hypotheses are tested by the researcher. It is only possible to statistically test a model that is overidentified (Stevens, 1996). This means that there are more pieces of information than unknowns. The unknown parameters are the parameters that were unequal to zero and needed to be estimated. These are the factor loadings, the covariances or correlations between the factors and the covariances or correlations between the measurement errors. The known pieces of information are the elements in the covariance matrix of the data. There are p(p+1) 2 unique pieces of information in this matrix, were p is the number of variables. According to Bollen (1989), in general each factor should at least have three variables to load on. During the estimation one tries to find the Λ, Φ and Φ δ that best reproduce Σ. However in general we do not have Σ, therefore the estimated matrix (ˆΣ = Σ(θ)) is compared to the sample covariance matrix. In R the default option that measures how close ˆΣ is to S is the maximum likelihood function. The maximum likelihood function assumes that the variables are continuous and multivariate normally distributed. Multivariate normality is defined in the following way (Mulaik, 2010): Definition. (Multivariate normal distribution) A p 1 vector x has a multivariate normal distribution with mean vector µ and covariance matrix Σ when its density function is given by: f(x) = 1 (2π) p/2 Σ 1/2 e( 1/2)(x µ)t Σ 1 (x µ)

20 THE UNDERLYING FACTORS OF DEPRESSION 20 Some properties of the multivariate normal distribution are: The contours of equal density for the multivariate normal distributions N(µ, Σ) are ellipsoids centered at µ. This ellipsoid is defined by the values of x such that (x µ) T Σ 1 (x µ) = c 2 ; Every linear combination of the components of the vector x has an univariate normal distribution; Every subset of x is multivariate normally distributed. Moreover each single component has a unit normal distribution. We will now derive the log likelihood function under the assumption that the data is multivariate normal distributed (Bollen, 1989): p(x) = n (2π) p 1 2 Σ(θ) 2 e i=1 l(θ) = np 2 log(2π) n 2 log Σ(θ) 1 2 ( ) (x i µ) T Σ 1 (θ)(x i µ) 2 n (x i µ) T Σ 1 (θ)(x i µ) = constant n 2 log Σ(θ) 1 n 2 tr[ (x i µ) T Σ 1 (θ)(x i µ)] i=1 i=1 = constant n 2 log Σ(θ) n 2 tr[s Σ 1 (θ)] (3) where S is an unbiased estimator of S. Moreover S = N 1 N S. This means that in large samples S and S are essentially the same. Moreover tr[s Σ 1 (θ)] is the trace of the matrix [S Σ 1 (θ)]. The maximum likelihood is equal to 2 n times the difference of the log likelihood of the model and the log likelihood of the saturated model. The saturated model is a model that predicts the data perfectly. So in that case the covariance matrix is precisely S. The difference in likelihood between the saturated model and the actual model is given by: F ML (S; Σ(θ)) = log Σ(θ) + tr(sσ 1 (θ)) log S tr(ss 1 ) = log Σ(θ) + tr(sσ 1 (θ)) log S tr(i) = log Σ(θ) + tr(sσ 1 (θ)) log S p. (4) Estimations of Λ, Φ and Φ δ are made in such way that F (S; ˆΣ) is as small as possible. The goal of maximum likelihood is to find parameters that maximizes the likelihood that the differences between S and ˆΣ result from sampling fluctuations (Stevens, 1996). To find these parameters one needs to minimize equation 4. This is equivalent to maximizing equation 3. Usually the maximum likelihood function can not be solved algebraically (Stevens, 1996). An iterative process is used in programs like R which estimates the parameters. The program tries to find a minimum of F (S; ˆΣ), by differ-

21 THE UNDERLYING FACTORS OF DEPRESSION 21 entiating this function and finding the roots. It starts with an initial value for ˆΣ and calculates the slope of F (S; ˆΣ) at these values. Subsequently the program will construct a tangent line at this point and will look for the roots of this tangent line. The roots of the tangent line is the new value for ˆΣ. The program continues with the same steps until it is converged. This is when the new ˆΣ is close enough to S or when no further improvements can be made within the constraints of the hypothesized model. This method is called the Newton-Raphson method (Upton & Cook, 2014). In figure 7 we see an example of a confirmatory factor analysis. There are 12 items and 2 factors namely cognitive and somatic. The curved arrow represents the factor correlation. The arrows from the factors to the items represent factor loadings. Moreover the arrows from the right into the items represent the measurement errors. Figure 7. : Example of a confirmatory factor analysis X1 X2 X3 X4 Cognitive X5 X6 X7 X8 Somatic X9 X10 X11 X12 Goodness of fit indices In section Exploratory factor analysis we already discussed the RMSEA as a goodness of fit index. There are however far more goodness of fit indices (Stevens, 1996). In this subsection we will discuss some of them. First of all the chi-square value is an

22 THE UNDERLYING FACTORS OF DEPRESSION 22 indication of the goodness of fit. It is however sensitive for sample size. A big sample size results in an inflated chi-square value (Brown, 2006). The chi-square value is calculated in the following way: χ 2 = 2F ML N 1. The degrees of freedom of chi-square test can be estimated by taking the number of known parameters and subtracting the number of unknown parameters. The known parameters are the unique elements in the covariance matrix. The unknown parameters are the loadings, error measurements, the factor variances and the factor correlations. The exact degrees of freedom are harder to calculate. They can be calculated in the following way (Claeskans & Hjort, 2009): first one takes the expected value of the second derivative of the log likelihood function, this is an s s matrix where s is the number of estimated parameters in the confirmatory factor analysis. We define this matrix as Q. Q = E( l xij (ˆθ)). Furthermore one needs to take the expected value of the inner product of the first derivative of the log likelihood function. This is also an s s matrix. P = E( l xij (ˆθ) l xij (ˆθ) T ). The true degrees of freedom are then calculated by taking the trace of the product of P and Q 1. However, in general it is very hard to find the matrices P and Q. Therefore one uses estimations of P and Q, respectively ˆP and ˆQ. These estimations are obtained by filling in all the data instead of just one data point. This method gives a good estimation, but has a lot of variance. Therefore the method of R is generally better. The Akaike Information Criterion (AIC) is an index that takes into account model fit as well as model complexity. AIC is often considered in evaluating competing, nonnested models (Brown, 2006). In general the model with the lowest AIC value is the best predicting model on average. AIC is computed with the following formula: AIC = 2 log likelihood + k npar where the log likelihood function (Equation 3) takes the natural logarithm of the likelihood function. The number of parameters in the fitted model is represented by npar. Moreover k is the penalty per used parameter; the default is k = 2. The Bayesian Information Criterion (BIC) uses k = log(n) (were n is the number of observations). Therefore BIC tends to favour smaller models than AIC does. Moreover BIC tends to favour the most correct model. Another class of fit indices are comparative fit indices (Brown, 2006). These indices are used for nested models and compare the fit of a specified model with a more restrictive baseline model. In general this baseline model sets all covariances of the

23 THE UNDERLYING FACTORS OF DEPRESSION 23 observed variables equal to zero. There are no constraints on the variances. One of the best behaving comparative fit indices is the comparative fit index (CFI). This index is computed in the following way: CF I = 1 max{χ 2 A df A, 0} max{χ 2 A df A, χ 2 B df B, 0} (5) where χ 2 A and df A are the chi-square value and the degrees of freedom of the target model. Moreover χ 2 B and df B are the chi-square value and the degrees of freedom of the baseline model. If max[χ 2 A df A, χ 2 B df B, 0] = 0, then the CFI can not be computed. The range of possible values of CFI is from 0 to 1. The closer the CFI is to 1, the better. Confirmatory factor analysis with ordinal data We assumed that all observed variables were continuous. However in psychological data this assumption is very often violated (e.g. Likert scale). We will call the vector x latent continuous indicators and x the observed variables. Recall that one can see data in the following way (for simplicity we do not denote the intercept): x = Λη + δ. However for categorical data we do not observe x, instead we observe a vector x with some or all categorical variables. Therefore we need a map that maps the ordinal values to the latent values. Furthermore the distributions of the observed variables differ from that for the latent continuous indicators. Moreover Acov(s ij, s gh ) Acov(s ij, s gh ), where s ij and s gh are elements of the covariance matrix for the observed variables x and s ij and s gh are elements of the covariance matrix for the latent continuous indicators x (Bollen, 1989). The asymptotic covariance matrix is an approximation to the covariance matrix of the sampling distribution of parameter estimates. This estimation gets better when the number of samples on which the parameter estimates are based increases. The most important violated assumption is for the population structure hypothesis. Assume that the population covariance matrix of the latent continuous indicators is equal to the population covariance matrix implied by θ, a vector with the model parameters. That is: Σ = Σ(θ). However in general the population covariance matrix of the latent continuous indicators does not equal the population covariance of the observed variables. Therefore Σ Σ(θ). Moreover Wylie (1976) concluded that the Pearson correlation coefficients between discrete variables are in general less than the correlation of the corresponding continuous variables. There is less difference between the correlations when the number of categories increases and the marginal distributions become similar. In that case Σ comes closer to Σ. The consequences of treating ordinal data as continuous data are quite serious. The chisquare test from the maximum likelihood function is negatively effected by immoderate kurtosis and skewness of the ordinal variables. Kurtosis and skewness seem to have a bigger influence on the chi-square test than the number of categories do (Bollen,

24 THE UNDERLYING FACTORS OF DEPRESSION ). A way of dealing with ordinal variables in confirmative factor analysis is using polychloric or polyserial correlations. Polychloric correlation estimates the linear relationship between two unobserved latent continuous variables, making use of the observed ordinal variables (Flora & Curran, 2004). Polyserial correlation estimates the linear relationship between two unobserved latent continuous variables, where one of the observed variables is continuous and the other is ordinal. Polychloric correlation relies on the assumption that the observed discrete values are due to underlying latent continuous variables. Moreover these pair of latent variables should have a bivariate normal distribution. A linear model relating x to η is not appropriate for ordinal variables. To correct this, we need a nonlinear function to relate the observed ordinal variables (x) to the latent continuous variables (x ). The relationship between x and x is the following: x i = c τ c 1 x i τ c where τ i is the category threshold, with c = 0,..., m i and m is the number of categories. Moreover τ 0 = and τ m =. We assume that the latent variable x is normally distributed, with density function φ(.) and distribution function Φ(.). Then the probability π c (i) of a response in category c on variable x i, is equal to (Yang-Wallentin, Jöreskog & Luo 2010) : (i) τ π c (i) c = P (x i = c) = τ (i) c 1 φ(u)du = Φ(τ c (i) ) Φ(τ (i) c 1 ) for c = 1,..., m i. The category thresholds can be computed in the following way: τ (i) c = Φ 1 (π (i) 1 + π (i) π (i) c ) where Φ 1 (x) is the inverse of the standardized normal distribution function and π (i) 1 + π(i) π(i) c is the probability of a response in category c or lower. To estimate π (i) c one can use the proportion of subjects in category c on variable x i c. Then the estimated thresholds (ˆτ i ) can be calculated with the following formula (Yang-Wallentin, Jöreskog & Luo, 2010): τˆ c = Φ 1 ( with c = 1,..., m 1 and N k is the number of subjects in the k th category and N is the total number of subjects. We will now introduce an example: c k=1 N k N ) Suppose, we asked 100 subjects how they have slept the past two weeks. There are four possible answers; 0, 1, 2 and 3, these answers mean respectively: I do not wake up at night, I have a restless and light sleep with a few brief awakenings each night, I wake up at least once a night, but I go back to sleep easily and I awaken more than once a night and stay awake

25 THE UNDERLYING FACTORS OF DEPRESSION 25 for 20 minutes or more, more than half the time. A table with the frequency of each option is shown below. Moreover we calculated the proportion and the cumulative proportion. Table 6: Categories and (cumulative) proportions Category Frequency Proportion Cumulative proportion So this results in the following estimations of the thresholds: Table 7: Thresholds threshold a 1 a 2 a 3 Estimate This means that when the latent continuous indicator x i is 0.50, then x i is in category 3. We still have to make a correction concerning the fact that the covariance structure hypothesis usually does not hold for discrete variables. This is where polychloric or polyserial correlation can be useful. We will concentrate on polychloric correlation. Suppose we have two ordinal variables x i and x j with respectively m i and m j categories. Moreover we assume that x i and x j are bivariate normally distributed with a mean of zero, unit variances and correlation ρ i,j. Maximum likelihood estimation is used to estimate the correlations between the latent continuous variables. The log likelihood is (Bollen, 1989): m j m i log L = A + i=1 j=1 N (ij) ab log(π (ij) ab ) where A is a constant, m i and m j are the number of categories for the first and second variable, N (ij) ab is the number of observations in category a of x i and category b of x j. Moreover π (ij) ab is defined in the following way (Yang-Wallentin, Jöreskog & Luo 2010): (i) τ π ij ab = P (x a i = a, x j = b) = τ (i) a 1 Moreover the density function is: τ (j) b τ (j) b 1 φ 2 (u, v)dudv. φ 2 = 1 2π 1 ρ 2 e 1 2(1 ρ 2 ) (u2 2ρuv+v 2 )

26 THE UNDERLYING FACTORS OF DEPRESSION 26 where τ ai and τ bj are the thresholds for respectively x j and y j and φ 2 is the bivariate normal distribution function with correlation ρ. To find the maximum of the log likelihood one needs to calculate partial derivatives with respect to ρ, a 1,..., a mi 1, b 1,..., b mj 1. The matrix with the polychloric correlations is a consistent estimator of Σ. A good estimation procedure to find Σ(θ) is the diagonally weighted least squares. This is the robust version of the weighted least squares function, which is defined as follows: F W LS (r, ρ, Σ(θ)) = (r ρ(θ)) T W 1 (r ρ(θ)) (6) where r is a p(p 1) 2 x 1 vector (p is the number of observed variables for x) with the polychloric correlations between all x i and x j. Moreover ρ(θ) is vector containing all elements of ΛΦΛ T below the diagonal. Finally W is a p(p 1) 2 x p(p 1) 2 positive definite matrix. Under very general assumptions, if the model holds in the population and if S converges in probability to Σ as the sample size increases, any fit function of the form of equation 6 with a positive definite W will give a consistent estimator of Λ and Φ (Jöreskog, 1994). However 2F ML is not exactly chi-square distributed. To obtain asymptotically correct chi-square measure of goodness of fit and asymptotically correct standard errors of parameter estimates one need to use a consistent estimator of the asymptotic covariance matrix ((N 1)Acov(r ij, r gh )) for W. By minimizing F W LS one can find Λ and Φ. So now one can test if Σ = Σ(θ). The diagonally weighted least squares estimation procedure takes W = diag(w ) (Yang-Wallentin, Jöreskog & Luo, 2010). A disadvantage of the weighted least square fitting function is that it requires the computation of an inverse of a p(p 1) 2 square matrix. Even with a moderate number of variables this can be quite a big matrix to invert. So for ordered confirmatory factor analysis the maximum likelihood function will not be computed. Therefore the calculations of some fit indices change. The RMSEA will be computed in a different way, namely: ( { χ 2 RMSEA = df max 1 }) n 1, 0 Measurement invariance One could be interested in how well measurement models generalise across different groups. To test this we can use measurement invariance. If a test is intended to be used in a heterogeneous population, then the test should have the same measurement properties across the different subgroups of the population (Brown, 2006). When a test does not measure the underlying constructs comparably across groups, the test is said to be biased. Biasness of a test can have serious consequences. For example when an IQ test is biased on genus, it will not represent the same true value of IQ for men and women. Measurement invariance is also used to measure invariance between repeated occasions with the same people. In R different kinds of measurement invari-