This Week

• Techniques
  • Confirmatory Factor Analysis (CFA)
• Key Functions
  • cfa( ) from lavaan package
• Reading
  • lavaan tutorial: http://lavaan.ugent.be/tutorial/tutorial.pdf (sections 1-4)
  • lavaan paper: https://www.jstatsoft.org/article/view/v048i02
  • Confirmatory Factor Analysis chapter on Learn

Learning Outcomes

• Know what it means to specify, estimate, and evaluate a CFA model
• Fit and interpret CFA models in R using the cfa( ) function
• Visualise CFA models using SEM diagrams

Overview of this lecture

• Introduction to CFA
• Model Specification
• Model Identification
• Model Estimation
• Model Evaluation
• Model Modification

Introduction to CFA

• Used to test a factor structure for a set of variables
• EFA is used when we have no fixed idea of our factor structure
• CFA is used to test a particular factor structure
• CFA tests how well our proposed factor structure fits the data
• Like EFA, CFA is a latent variable model
  • observed variables serve as indicators of underlying latent factors
• Unlike EFA, only specific loadings are included in the model

The variance-covariance matrix

• Our starting point for CFA is the variance-covariance matrix for our items
• CFA models represent these variances/covariances in terms of a set of latent factors

round(cov(agg.items),2)

##        item1 item2 item3 item4 item5 item6 item7 item8 item9 item10
## item1   0.98  0.56  0.47  0.44  0.59 -0.01  0.09  0.06  0.10   0.05
## item2   0.56  1.00  0.53  0.52  0.67  0.01  0.10  0.06  0.11   0.06
## item3   0.47  0.53  0.98  0.46  0.58  0.04  0.09  0.04  0.10   0.02
## item4   0.44  0.52  0.46  1.00  0.56  0.04  0.10  0.04  0.11   0.06
## item5   0.59  0.67  0.58  0.56  1.01  0.01  0.08  0.04  0.08   0.04
## item6  -0.01  0.01  0.04  0.04  0.01  1.03  0.58  0.60  0.43   0.45
## item7   0.09  0.10  0.09  0.10  0.08  0.58  0.98  0.81  0.58   0.60
## item8   0.06  0.06  0.04  0.04  0.04  0.60  0.81  1.02  0.61   0.64
## item9   0.10  0.11  0.10  0.11  0.08  0.43  0.58  0.61  0.98   0.45
## item10  0.05  0.06  0.02  0.06  0.04  0.45  0.60  0.64  0.45   1.01

SEM Diagrams

Exogenous versus endogeneous variables

• exogenous variables receive input from no other variables
  • they emanate but are not on the end of single-headed arrow paths
  • they are the ‘independent variables’ or ‘predictors’
• endogenous variables receive input from other variables
  • they are on the end of single-headed arrow paths
  • they are the ‘dependent variables’ or ‘outcomes’
  • they may also be predictors of other variables in the model

SEM diagram for a simple regression model

SEM diagram for a covariance

SEM diagram for a path analysis model

SEM diagram for another path analysis model

SEM diagram for a more complex model

BREAK 1

• Time for a pause
• Quiz question:
  • Which of these are differences between an EFA and a CFA:
    • 1. EFA only involves observed variables but CFA involves both observed and latent variables
    • 2. EFA estimates all possible loadings but CFA usually only estimates some
    • 3. EFA can include multiple latent variables but CFA can only include one
    • 4. CFA tests causality while EFA can only test correlation

Welcome back 1

• Welcome back!

• The answer to the quiz question is…

  • 1. EFA only involves observed variables but CFA involves both observed and latent variables
  • 2. EFA estimates all possible loadings but CFA usually only estimates some
  • 3. EFA can include multiple latent variables but CFA can only include one
  • 4. CFA tests causality while EFA can only test correlation

The CFA model

The parameters of a CFA

• Latent factor variances and covariances
  • The variability of and associations between the latent factors
• Factor loadings
  • Regression of the latent variables on the observed variables
  • Strength of relation between underlying latent variables and observed variables
• Residual variances
  • Variance in the observed variables not explained by the latent variables
• Residual covariances
  • The covariances between observed variables that exist over and above their covariance due to their shared relation with a latent factor
• CFAs involve finding (or specifying) values for all of these parameters

Model specification

• Defining the model we want to test
  • i.e., which parameters do we want to estimate?
    • How many factors?
    • Which items do we think go with which factors?
    • Are the factors correlated?
• Based on theory or past research (e.g., previous EFAs)

Latent variable scaling

• Latent variable scaling is a key aspect of model specification
• Latent variables have no inherent scale, so we have to define one
• Two commonly used scaling methods:
  • Method 1: Fix the variance of each latent variable to 1
  • Method 2: Fix one factor loading for each latent variable to 1
• Note that the necessity of scaling also applies to the residual factors
  • Typically uses Method 2

Scaling the latent variables by fixing factor variances

Scaling the latent variables by fixing factor loadings

Model identification

• More generally, we need to ensure that the model we specify is identified
• Identification concerns the number of ‘knowns’ versus ‘unknowns’
• There must be more knowns than unknowns in order to be able to test a CFA
• In CFA, the knowns are variances and covariances of the observed variables
• The unknowns are the parameters we want to estimate
• Degrees of freedom are the difference between knowns and unknowns

Levels of identification

• There are three levels of identification:
• Under-identified models: have < 0 degrees of freedom
• Just Identified models: have 0 degrees of freedom
• Over-Identified models: have > 0 degrees of freedom

Model identification illustration

• Chou & Bentler (1995) provide an illustration based on simultaneous linear equations:
  • Eq.1: \(x + y = 5\)
  • Eq.2: \(2x + y = 8\)
  • Eq.3: \(x + 2y = 9\)
• Eq.1 is on its own is under-identified
• Eq.1 & 2 are together just identified
• Eq.1, 2 & 3 are together over identified

The number of knowns

• To ensure model identification, we need to know the number of knowns

• We can calculate the knowns by:

  \[ \frac{\left(k+1 \right)\left(k \right)}{2} \]

• where k is the number of observed variables.

The number of knowns

• This is the number of unique elements in the variance-covariance matrix for our observed variables
  • e.g., if we had three observed variables:

round(cov(Three.variables),2)

##      V1   V2   V3
## V1 1.05 0.33 0.42
## V2 0.33 1.03 0.65
## V3 0.42 0.65 1.01

• We have 6 unique elements (3 variances and 3 covariances)

Implications for CFA

• We usually need a minimum of three observed variables for a just identified model

BREAK 2

• Time for a pause
• Quiz question:
  • A CFA model with 3 degrees of freedom would be best described as:
    • Under-identified
    • Over-identified
    • Just identified
    • Negatively identified

Welcome back 2

• Welcome back!
• The answer to the quiz question is:
  • A CFA model with 3 degrees of freedom would be best described as:
    • Under-identified
    • Over-identified
    • Just identified
    • Negatively identified

Model estimation

• After we have specified our model (& checked it is identified) we proceed to estimation
• Model estimation refers to finding the ‘best’ values for the unknown parameters

Specifying which parameters to estimate…

Finding the parameter values

Maximum likelihood estimation

• Maximum likelihood estimation is most commonly used
• Finds the parameters that maximise the likelihood of the data
• Begins with a set of starting values
• Iterative process of improving these values
  • i.e. to minimise the difference between the sample covariance matrix and the covariance matrix implied by the parameter values
• Terminates when the values are no longer substantially improved across iterations
  • At this point convergence is said to have been reached

No convergence?

• Sometimes estimation fails
• Common reasons are:
  • The model is not identified
  • The model is very mis-specified
  • The model is very complex so more iterations are needed than the program default

Maximum likelihood estimation assumptions

• Large sample size
• Multivariate normality
• Variables are on a continuous scale

Alternative estimators

• Robust maximum likelihood estimation
  • For non-normal data
• Weighted least squares, unweighted least squares or diagonally weighted least squares
  • For ordinal data

BREAK 3

• Time for a pause
• Quiz question:
  • Which of these is most likely to result in a model failing to converge using maximum likelihood estimation:
    • Model under-identification
    • Skewed variables
    • A sample size of only n=100
    • Ordinal variables with 5 response options

Welcome back 3

• Welcome back!
• The answer to the quiz question is…
  • Which of these is most likely to result in a model failing to converge using maximum likelihood estimation:
    • Model under-identification
    • Skewed variables
    • A sample size of only n=100
    • Ordinal variables with 5 response options

Model evaluation

• Once the model has been evaluated, we ask: how good is the model?
  • Global fit
  • Local fit

Global fit

• \(\chi^2\)
  • When we use maximum likelihood estimation we obtain a \(\chi^2\) value for the model
  • This can be compared to a \(\chi^2\) distribution with degrees of freedom equal to our model degrees of freedom
  • Statistically significant \(\chi^2\) suggests the CFA model does not do a good job of reproducing the observed variance-covariance matrix
    
• However, \(\chi^2\) does not work well in practice
  • Leads to the rejection of models that are only trivially mis-specified

Alternatives to \(\chi^2\)

• Absolute fit
  • Standardised root mean square residual (SRMR)
    • measures the discrepancy between the observed correlation matrix and model-implied correlation matrix
    • ranges from 0 to 1 with 0=perfect fit
    • values <.05 considered good
• Parsimony-corrected
  • Corrects for the complexity of the model
  • Adds a penalty for having more degrees of freedom
  • Root mean square square error of approximation (RMSEA)
    • 0=perfect fit
    • values <.05 considered good

Incremental fit indices

• Compares the model to a more restricted baseline model
  • Usually an ‘independence’ model where all observed variable covariances fixed to 0
• Comparative fit index (CFI)
  • ranges between 0 and 1 with 1=perfect fit
  • values > .95 considered good
• Tucker-Lewis index (TLI)
  • includes a parsimony penalty
  • values >.95 considered good

Local fit

• It is also possible to examine local areas of mis-fit
• Modification indices estimate the improvement in \(\chi^2\) that could be expected from including an additional parameter
  • e.g., a cross-loading, residual covariance or latent variable covariance
• Expected parameter changes estimate the value of the parameter were it to be included

Making model modifications

• Modification indices and expected parameter changes can be helpful for identifying how to improve the model
• However:
  • Modifications should be made iteratively
  • Don’t go overboard: may just be capitalising on chance
  • Make sure the modifications can be justified on substantive grounds
  • Be aware that this becomes an exploratory modeling practice
  • Ideally replicate the new model in an independent sample

Other considerations in model evaluation

• Ideally:
  • Factor loadings should be statistically significant
  • Standardised factor loadings should be >|.30|
  • Else some items/parameters could be trimmed from the model
  • (with the same caveats as on previous slide)
• Check for Heywood cases
  • Parameter estimates that are outside the permissable range
  • E.g., correlations >1, negative residual variances
  • May require modifications to the model to address

BREAK 4

• Time for a pause
• Quiz question
• Which of these fit indices compares a CFA model fit to an ‘independence model’ fit:
  • Comparative fit index (CFI)
  • Root Mean Square Error of Approximation (RMSEA)
  • Standardised Root Mean Square Residual (SRMR)
  • Chi-square

Welcome back 4

• Welcome back!
• Which of these fit indices compares a CFA model fit to an ‘independence model’ fit:
  • Comparative fit index (CFI)
  • Root Mean Square Error of Approximation (RMSEA)
  • Standardised Root Mean Square Residual (SRMR)
  • Chi-square

Interpreting a CFA

• To aid interpretation we can request a fully standardised solution
• Converts loadings/covariances to a correlation metric
• Thereafter, the interpretation is similar to EFA:
  • Loadings tell us strength of association between latent factor and items
  • Factor correlations tell us how strongly associated latent factors are

Conducting a CFA model using lavaan

• Lavaan = Latent Variable Analysis
• Used to specify and estimate latent variable models
• Three steps:

#step 1: specify the model
 
model<-'LV=~V1+V2+V3+V4'
  # we write the model using lavaan syntax enclosed in single quote marks

#step2: estimate the model

model.est<-cfa(model=model, data=df)  
   # 'model= ' refers to a lavaan syntax object with the model specification
   # 'data= ' gives name of the dataframe in which to find the variables
#step3: inspect the results

summary(model.est)
    # the summary function shows us output from a fitted model

Model specification

• Specification uses lavaan syntax:

# simple regression model 

Regression<-'DV~IV'

# multiple regression model

Multiple.regression<-'DV~IV1+IV2+IV3'

#covariance between two variables

Covariance<-'V1~~V2'

#latent factor specification

CFA<-'LV=~V1+V2+V3+V4'

Model specification for our aggression example

1. I hit someone
2. I kicked someone
3. I shoved someone
4. I battered someone
5. I physically hurt someone on purpose
6. I deliberately insulted someone
7. I swore at someone
8. I threatened to hurt someone
9. I called someone a nasty name to their face
10. I shouted mean things at someone

agg_m<-
  'Pagg=~item1+item2+item3+item4+item5
  
   Vagg=~item6+item7+item8+item9+item10
   
   Pagg~~Vagg'

Model estimation in lavaan

• To estimate the model, we then feed the object we just created into the cfa( ) function
• We also name the dataset containing the model
  • Lavaan will compute the variance-covariance matrix internally

agg_m.est<-cfa(agg_m, data=agg.items)

Scaling constraints

• By default, cfa( ) will scale the latent variables by fixing the first indicator for each latent factor to 1
• To override this and fix latent factor variances instead, we can write:

agg_m.est<-cfa(agg_m, data=agg.items, std.lv=T)

Model evaluation

• We can check the model fit using the summary( ) function:

summary(agg_m.est, fit.measures=T)

## lavaan 0.6-5 ended normally after 23 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                         21
##                                                       
##   Number of observations                          1000
##                                                       
## Model Test User Model:
##                                                       
##   Test statistic                                41.739
##   Degrees of freedom                                34
##   P-value (Chi-square)                           0.170
## 
## Model Test Baseline Model:
## 
##   Test statistic                              4711.354
##   Degrees of freedom                                45
##   P-value                                        0.000
## 
## User Model versus Baseline Model:
## 
##   Comparative Fit Index (CFI)                    0.998
##   Tucker-Lewis Index (TLI)                       0.998
## 
## Loglikelihood and Information Criteria:
## 
##   Loglikelihood user model (H0)             -11838.328
##   Loglikelihood unrestricted model (H1)     -11817.459
##                                                       
##   Akaike (AIC)                               23718.657
##   Bayesian (BIC)                             23821.720
##   Sample-size adjusted Bayesian (BIC)        23755.023
## 
## Root Mean Square Error of Approximation:
## 
##   RMSEA                                          0.015
##   90 Percent confidence interval - lower         0.000
##   90 Percent confidence interval - upper         0.029
##   P-value RMSEA <= 0.05                          1.000
## 
## Standardized Root Mean Square Residual:
## 
##   SRMR                                           0.024
## 
## Parameter Estimates:
## 
##   Information                                 Expected
##   Information saturated (h1) model          Structured
##   Standard errors                             Standard
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   Pagg =~                                             
##     item1             0.695    0.029   24.157    0.000
##     item2             0.792    0.028   28.463    0.000
##     item3             0.678    0.029   23.387    0.000
##     item4             0.656    0.030   22.003    0.000
##     item5             0.850    0.027   31.231    0.000
##   Vagg =~                                             
##     item6             0.652    0.030   21.967    0.000
##     item7             0.873    0.025   34.276    0.000
##     item8             0.922    0.025   36.289    0.000
##     item9             0.662    0.029   23.157    0.000
##     item10            0.691    0.029   24.006    0.000
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   Pagg ~~                                             
##     Vagg              0.098    0.035    2.792    0.005
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .item1             0.491    0.026   19.051    0.000
##    .item2             0.369    0.022   16.488    0.000
##    .item3             0.515    0.027   19.360    0.000
##    .item4             0.572    0.029   19.844    0.000
##    .item5             0.286    0.021   13.748    0.000
##    .item6             0.605    0.029   20.891    0.000
##    .item7             0.219    0.016   13.978    0.000
##    .item8             0.168    0.015   10.988    0.000
##    .item9             0.541    0.026   20.657    0.000
##    .item10            0.532    0.026   20.469    0.000
##     Pagg              1.000                           
##     Vagg              1.000

Model evaluation

• We can examine local mis-specifications using the modindices() function

modindices(agg_m.est, sort=T)

##      lhs op    rhs    mi    epc sepc.lv sepc.all sepc.nox
## 26  Pagg =~  item8 7.281 -0.051  -0.051   -0.051   -0.051
## 27  Pagg =~  item9 7.099  0.069   0.069    0.070    0.070
## 25  Pagg =~  item7 6.654  0.050   0.050    0.050    0.050
## 38 item1 ~~  item6 5.283 -0.044  -0.044   -0.080   -0.080
## 61 item4 ~~  item8 4.760 -0.030  -0.030   -0.096   -0.096
## 24  Pagg =~  item6 3.288 -0.050  -0.050   -0.049   -0.049
## 57 item3 ~~ item10 2.572 -0.029  -0.029   -0.056   -0.056
## 60 item4 ~~  item7 2.231  0.021   0.021    0.059    0.059
## 33  Vagg =~  item5 2.018 -0.032  -0.032   -0.032   -0.032
## 36 item1 ~~  item4 1.997 -0.028  -0.028   -0.054   -0.054
## 53 item3 ~~  item6 1.701  0.025   0.025    0.045    0.045
## 40 item1 ~~  item8 1.513  0.016   0.016    0.055    0.055
## 62 item4 ~~  item9 1.427  0.023   0.023    0.041    0.041
## 46 item2 ~~  item6 1.333 -0.020  -0.020   -0.043   -0.043
## 56 item3 ~~  item9 1.290  0.021   0.021    0.040    0.040
## 69 item6 ~~  item7 1.050  0.017   0.017    0.046    0.046
## 59 item4 ~~  item6 0.929  0.019   0.019    0.033    0.033
## 49 item2 ~~  item9 0.816  0.015   0.015    0.033    0.033
## 47 item2 ~~  item7 0.815  0.011   0.011    0.038    0.038
## 70 item6 ~~  item8 0.760 -0.015  -0.015   -0.046   -0.046
## 55 item3 ~~  item8 0.730 -0.011  -0.011   -0.038   -0.038
## 34 item1 ~~  item2 0.698  0.016   0.016    0.038    0.038
## 77 item8 ~~ item10 0.678  0.014   0.014    0.046    0.046
## 32  Vagg =~  item4 0.642  0.021   0.021    0.021    0.021
## 51 item3 ~~  item4 0.615  0.016   0.016    0.029    0.029
## 67 item5 ~~  item9 0.611 -0.012  -0.012   -0.031   -0.031
## 75 item7 ~~ item10 0.538 -0.012  -0.012   -0.035   -0.035
## 48 item2 ~~  item8 0.524 -0.008  -0.008   -0.034   -0.034
## 54 item3 ~~  item7 0.503  0.009   0.009    0.028    0.028
## 45 item2 ~~  item5 0.294 -0.012  -0.012   -0.036   -0.036
## 76 item8 ~~  item9 0.275  0.009   0.009    0.028    0.028
## 41 item1 ~~  item9 0.262  0.009   0.009    0.018    0.018
## 43 item2 ~~  item3 0.246 -0.010  -0.010   -0.022   -0.022
## 30  Vagg =~  item2 0.229  0.011   0.011    0.011    0.011
## 50 item2 ~~ item10 0.214  0.008   0.008    0.017    0.017
## 37 item1 ~~  item5 0.182  0.008   0.008    0.023    0.023
## 29  Vagg =~  item1 0.164  0.010   0.010    0.010    0.010
## 28  Pagg =~ item10 0.161 -0.010  -0.010   -0.010   -0.010
## 74 item7 ~~  item9 0.133 -0.006  -0.006   -0.017   -0.017
## 63 item4 ~~ item10 0.123  0.007   0.007    0.012    0.012
## 78 item9 ~~ item10 0.107 -0.006  -0.006   -0.011   -0.011
## 65 item5 ~~  item7 0.094 -0.004  -0.004   -0.014   -0.014
## 58 item4 ~~  item5 0.076  0.005   0.005    0.013    0.013
## 31  Vagg =~  item3 0.065  0.007   0.007    0.007    0.007
## 35 item1 ~~  item3 0.055 -0.005  -0.005   -0.009   -0.009
## 71 item6 ~~  item9 0.053 -0.005  -0.005   -0.008   -0.008
## 68 item5 ~~ item10 0.051  0.004   0.004    0.009    0.009
## 72 item6 ~~ item10 0.048  0.004   0.004    0.008    0.008
## 44 item2 ~~  item4 0.038  0.004   0.004    0.008    0.008
## 73 item7 ~~  item8 0.034 -0.004  -0.004   -0.022   -0.022
## 64 item5 ~~  item6 0.021 -0.002  -0.002   -0.006   -0.006
## 66 item5 ~~  item8 0.020 -0.002  -0.002   -0.007   -0.007
## 39 item1 ~~  item7 0.020 -0.002  -0.002   -0.006   -0.006
## 42 item1 ~~ item10 0.005 -0.001  -0.001   -0.003   -0.003
## 52 item3 ~~  item5 0.003  0.001   0.001    0.003    0.003

Standardised parameter estimates

• We can also inspect the standardised parameter estimates

summary(agg_m.est, standardized=T)

## lavaan 0.6-5 ended normally after 23 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                         21
##                                                       
##   Number of observations                          1000
##                                                       
## Model Test User Model:
##                                                       
##   Test statistic                                41.739
##   Degrees of freedom                                34
##   P-value (Chi-square)                           0.170
## 
## Parameter Estimates:
## 
##   Information                                 Expected
##   Information saturated (h1) model          Structured
##   Standard errors                             Standard
## 
## Latent Variables:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Pagg =~                                                               
##     item1             0.695    0.029   24.157    0.000    0.695    0.704
##     item2             0.792    0.028   28.463    0.000    0.792    0.793
##     item3             0.678    0.029   23.387    0.000    0.678    0.687
##     item4             0.656    0.030   22.003    0.000    0.656    0.655
##     item5             0.850    0.027   31.231    0.000    0.850    0.846
##   Vagg =~                                                               
##     item6             0.652    0.030   21.967    0.000    0.652    0.642
##     item7             0.873    0.025   34.276    0.000    0.873    0.881
##     item8             0.922    0.025   36.289    0.000    0.922    0.914
##     item9             0.662    0.029   23.157    0.000    0.662    0.669
##     item10            0.691    0.029   24.006    0.000    0.691    0.688
## 
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   Pagg ~~                                                               
##     Vagg              0.098    0.035    2.792    0.005    0.098    0.098
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##    .item1             0.491    0.026   19.051    0.000    0.491    0.504
##    .item2             0.369    0.022   16.488    0.000    0.369    0.371
##    .item3             0.515    0.027   19.360    0.000    0.515    0.528
##    .item4             0.572    0.029   19.844    0.000    0.572    0.570
##    .item5             0.286    0.021   13.748    0.000    0.286    0.284
##    .item6             0.605    0.029   20.891    0.000    0.605    0.588
##    .item7             0.219    0.016   13.978    0.000    0.219    0.223
##    .item8             0.168    0.015   10.988    0.000    0.168    0.165
##    .item9             0.541    0.026   20.657    0.000    0.541    0.552
##    .item10            0.532    0.026   20.469    0.000    0.532    0.527
##     Pagg              1.000                               1.000    1.000
##     Vagg              1.000                               1.000    1.000

Visualising the model

• Sempaths() from the semPlot package can be used to visual a model as a SEM diagram

semPaths(agg_m.est, what='stand')

Writing up a CFA model

• Methods
  • Model(s) being tested
  • Scaling /identifcation method
  • Estimation method
  • Criteria that used to judge fit
• Results
  • Model fit (\(\chi^2\) test, CFI, TLI, RMSEA, SRMR)
  • Any modifications made and why
  • Model parameters (in a SEM diagram or table)

Cautions regarding CFA

• Good fit doesn’t guarantee that the model is ‘correct’
• Be careful about ‘reifying’ latent variables
• Even when there are no common factors, CFA models can fit well

Summary

CFA involves testing a hypothesised factor structure

• Specifying a model
  • Identification and scaling
• Estimating that model
  • e.g., maximum likelihood estimation
• Seeing how well that model fits the data
  • Global and local fit
• Interpreting the model