SEM, Revealed.

# SEM, Revealed.

## Modeling connections with latent variables and regression pathways.

###

Slides available at <https://tinyurl.com/CHBD-slides24>
 
PDF slides at <https://tinyurl.com/CHBD-pdf24>

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## Waylon Howard | Biostatistician @ BEAR/CHBD

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was founded with a collaborative spirit and lofty objective: to deliver comprehensive data management, advanced analytics, and expert statistical, epidemiologic, and qualitative support…

By alleviating external consultants, designing studies in-house, and engaging investigators directly, we’re able to provide responsive, efficient, and high-quality data support at a fraction of the going cost.

???
Notes here

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## Biostatistics @ BEAR

Supports stakeholders throughout SCRI by helping them make better decisions using data

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- [How to measure it?](#measure)    
- [Fitting a CFA model.](#cfa)    
- [Estimators.](#ml)    
- [Model Fit.](#fit)    
- [Statistical Code.](#software)
- [Example SEM Models.](#results)
- [Power.](#results)

---

<div class="my-footer">Slides at tinyurl.com/CHBD-slides24 &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; &emsp; Return to <a style="color:White;" href="slide_deck.html#toc">Topic List</a></div>

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# Perceived Social Support
## How to measure it?

---

## Self-report questions

***

1. My friends really try to help me.
1. I can count on my friends when things go wrong.
1. I can talk about my problems with my friends.

***

<table class="table" style="margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:center;"> Very Strongly Disagree </th>
 <th style="text-align:center;"> Strongly Disagree </th>
 <th style="text-align:center;"> Mildly Disagree </th>
 <th style="text-align:center;"> Neutral </th>
 <th style="text-align:center;"> Mildly Agree </th>
 <th style="text-align:center;"> Strongly Agree </th>
 <th style="text-align:center;"> Very Strongly Agree </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:center;"> 1 </td>
 <td style="text-align:center;"> 2 </td>
 <td style="text-align:center;"> 3 </td>
 <td style="text-align:center;"> 4 </td>
 <td style="text-align:center;"> 5 </td>
 <td style="text-align:center;"> 6 </td>
 <td style="text-align:center;"> 7 </td>
 </tr>
</tbody>
</table>

<center><medium> Higher scores = More Perceived Social Support </medium></center>

---

1. My friends really try to help me.

***

`\(X_i\)` = `\(T_i\)` + ( `\(S_i\)` + `\(e_i\)` )

`\(T_i\)` is the 'true' score

`\(S_i\)` is item-specific, yet reliable

`\(e_i\)` is random error, or noise

]

**Use the scoring procedure**: No measurement error (*always perfectly measured*), uniform effectiveness of items (*equal items*), invariance across groups and time.
 
]

---

---

`\(X_i\)` = `\(T_i\)` + ( `\(S_i\)` + `\(e_i\)` )

---

# How do we get the true score? 
## Fitting a CFA model.

---

Estimated Parameters: **7** Observed Information: **6**

]

**Matrix Formula:** `\(\Sigma = \Lambda \Psi \Lambda'+ \Theta\)`

Σ = **Variance/Covariance Matrix**

**Implied Variance/Covariance Matrix**

]

*Underidentified*: `\(X + Y = 20\)`

---

Variance /Covariance Matrix

**Just Identified.**

]

Fix the latent variance to 1.0

<img class="rectangle" src="images/setscale1.png" width="275px"/>
 
]

---

Variance /Covariance Matrix

**Just Identified.**

]

Fix the loading to 1.0

<img class="rectangle" src="images/setscale2.png" width="275px"/>
 
]

---

Variance /Covariance Matrix

**Just Identified.**

]

Constrain loading to average 1.0

<img class="rectangle" src="images/setscale3.png" width="325px"/>
 
]

---

Variance /Covariance Matrix

]

<img class="rectangle" src="images/cfa6.png" width="275px"/>
 
]

---

# How do we get the numbers? 
## Estimators.

---

## Maximum Likelihood

`\(L_i\)` = `\(\frac {1}{\sqrt{2\pi\sigma^2}}\)` e[-.5 `\(\frac{(Y_i - \mu)^2}{\sigma^2}\)` ]

> ML identifies the population parameters that are most likely given the observed data 
> A likelihood (or log likelihood) function quantifies the fit of the data to the parameters 
> ML requires a population distribution (normal)

---

`\(L_i\)` = `\(\frac {1}{\sqrt{2\pi\sigma^2}}\)` e[-.5 `\(\frac{(Y_i - \mu)^2}{\sigma^2}\)` ]

Applying the density function gives the relative probability `\((L_i)\)` of each score from this normal distribution.

Score `\((\mu\)` = 50.44, `\(\sigma\)` = 58.68 `\()\)`

]

<table class="table" style="font-size: 12px; margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:center;"> Person ID </th>
 <th style="text-align:center;"> Score </th>
 <th style="text-align:center;"> Likelihood </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:center;"> 1 </td>
 <td style="text-align:center;"> 36.6 </td>
 <td style="text-align:center;"> 0.00661212 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 2 </td>
 <td style="text-align:center;"> 41.8 </td>
 <td style="text-align:center;"> 0.006725313 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 3 </td>
 <td style="text-align:center;"> 42.6 </td>
 <td style="text-align:center;"> 0.006738201 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 4 </td>
 <td style="text-align:center;"> 43.1 </td>
 <td style="text-align:center;"> 0.006745631 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 5 </td>
 <td style="text-align:center;"> 43.4 </td>
 <td style="text-align:center;"> 0.006749858 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 6 </td>
 <td style="text-align:center;"> 44.2 </td>
 <td style="text-align:center;"> 0.006760279 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 7 </td>
 <td style="text-align:center;"> 44.9 </td>
 <td style="text-align:center;"> 0.006768379 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 8 </td>
 <td style="text-align:center;"> 46.3 </td>
 <td style="text-align:center;"> 0.006781711 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 9 </td>
 <td style="text-align:center;"> 48.6 </td>
 <td style="text-align:center;"> 0.006795269 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 10 </td>
 <td style="text-align:center;"> 49.0 </td>
 <td style="text-align:center;"> 0.006796563 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 11 </td>
 <td style="text-align:center;"> 50.0 </td>
 <td style="text-align:center;"> 0.006798419 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 12 </td>
 <td style="text-align:center;"> 51.6 </td>
 <td style="text-align:center;"> 0.006797282 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 13 </td>
 <td style="text-align:center;"> 54.6 </td>
 <td style="text-align:center;"> 0.006781547 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 14 </td>
 <td style="text-align:center;"> 54.8 </td>
 <td style="text-align:center;"> 0.00677987 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 15 </td>
 <td style="text-align:center;"> 55.7 </td>
 <td style="text-align:center;"> 0.006771351 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 16 </td>
 <td style="text-align:center;"> 57.2 </td>
 <td style="text-align:center;"> 0.006753646 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 17 </td>
 <td style="text-align:center;"> 57.6 </td>
 <td style="text-align:center;"> 0.006748188 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 18 </td>
 <td style="text-align:center;"> 60.3 </td>
 <td style="text-align:center;"> 0.006703308 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 19 </td>
 <td style="text-align:center;"> 60.9 </td>
 <td style="text-align:center;"> 0.006691451 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 20 </td>
 <td style="text-align:center;"> 65.3 </td>
 <td style="text-align:center;"> 0.006584073 </td>
 </tr>
</tbody>
</table>

]

<center><medium> Largest relative probability is for ID 11 </medium></center>

---

---

Multiply each `\(L_i\)` to get sample likelihood. 
0.0000000000000000000000000000000000000000000378615

To avoid small numbers, we add the log of the likelihood. 
-99.9824
]

<table class="table" style="font-size: 12px; margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:center;"> Person ID </th>
 <th style="text-align:center;"> HRQoL </th>
 <th style="text-align:center;"> Likelihood </th>
 <th style="text-align:center;"> LogLikelihood </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:center;"> 1 </td>
 <td style="text-align:center;"> 36.6 </td>
 <td style="text-align:center;"> 0.00661212 </td>
 <td style="text-align:center;"> -5.0189 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 2 </td>
 <td style="text-align:center;"> 41.8 </td>
 <td style="text-align:center;"> 0.006725313 </td>
 <td style="text-align:center;"> -5.0019 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 3 </td>
 <td style="text-align:center;"> 42.6 </td>
 <td style="text-align:center;"> 0.006738201 </td>
 <td style="text-align:center;"> -5.0000 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 4 </td>
 <td style="text-align:center;"> 43.1 </td>
 <td style="text-align:center;"> 0.006745631 </td>
 <td style="text-align:center;"> -4.9989 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 5 </td>
 <td style="text-align:center;"> 43.4 </td>
 <td style="text-align:center;"> 0.006749858 </td>
 <td style="text-align:center;"> -4.9982 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 6 </td>
 <td style="text-align:center;"> 44.2 </td>
 <td style="text-align:center;"> 0.006760279 </td>
 <td style="text-align:center;"> -4.9967 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 7 </td>
 <td style="text-align:center;"> 44.9 </td>
 <td style="text-align:center;"> 0.006768379 </td>
 <td style="text-align:center;"> -4.9955 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 8 </td>
 <td style="text-align:center;"> 46.3 </td>
 <td style="text-align:center;"> 0.006781711 </td>
 <td style="text-align:center;"> -4.9935 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 9 </td>
 <td style="text-align:center;"> 48.6 </td>
 <td style="text-align:center;"> 0.006795269 </td>
 <td style="text-align:center;"> -4.9915 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 10 </td>
 <td style="text-align:center;"> 49.0 </td>
 <td style="text-align:center;"> 0.006796563 </td>
 <td style="text-align:center;"> -4.9913 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 11 </td>
 <td style="text-align:center;"> 50.0 </td>
 <td style="text-align:center;"> 0.006798419 </td>
 <td style="text-align:center;"> -4.9911 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 12 </td>
 <td style="text-align:center;"> 51.6 </td>
 <td style="text-align:center;"> 0.006797282 </td>
 <td style="text-align:center;"> -4.9912 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 13 </td>
 <td style="text-align:center;"> 54.6 </td>
 <td style="text-align:center;"> 0.006781547 </td>
 <td style="text-align:center;"> -4.9935 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 14 </td>
 <td style="text-align:center;"> 54.8 </td>
 <td style="text-align:center;"> 0.00677987 </td>
 <td style="text-align:center;"> -4.9938 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 15 </td>
 <td style="text-align:center;"> 55.7 </td>
 <td style="text-align:center;"> 0.006771351 </td>
 <td style="text-align:center;"> -4.9951 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 16 </td>
 <td style="text-align:center;"> 57.2 </td>
 <td style="text-align:center;"> 0.006753646 </td>
 <td style="text-align:center;"> -4.9977 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 17 </td>
 <td style="text-align:center;"> 57.6 </td>
 <td style="text-align:center;"> 0.006748188 </td>
 <td style="text-align:center;"> -4.9985 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 18 </td>
 <td style="text-align:center;"> 60.3 </td>
 <td style="text-align:center;"> 0.006703308 </td>
 <td style="text-align:center;"> -5.0052 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 19 </td>
 <td style="text-align:center;"> 60.9 </td>
 <td style="text-align:center;"> 0.006691451 </td>
 <td style="text-align:center;"> -5.0069 </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 20 </td>
 <td style="text-align:center;"> 65.3 </td>
 <td style="text-align:center;"> 0.006584073 </td>
 <td style="text-align:center;"> -5.0231 </td>
 </tr>
</tbody>
</table>

]

<center><medium> Largest relative probability is for ID 11 </medium></center>

---

---

## Model Fit

<medium> Your data = Model-implied? </medium> 

Chi-square `\((\chi^2)\)` = -2 * (Null Loglikelihood - Alternative Loglikelihood)

---

## Absolute Model Fit: 
*How far from perfect*:

**RMSEA**, **SRMR**

]

> `\(>\)` **.10** poor fit 
> **.08 - .10** mediocre fit 
> **.05 - .08** acceptable fit 
> **.01 - .05** close fit 
> **.00** exact fit

]

---

## Relative Model Fit: 
*How far from worst*:

**TLI (NNFI)**, **CFI**...

]

> `\(<\)` **.85** poor fit 
> **.85-.90** mediocre fit 
> **.90-.95**	acceptable fit 
> **.95-.99** close fit 
> **1.00** exact fit

]

Also: Modification indices, Fitted residual matrix, Parameter estimates…

---

]

Chi-Square = -2[(-1365.848) - (-1351.359)] = **28.978** 
DF = `\(\frac{v(v+1)}{2}-p\)` = `\(\frac{6(6+1)}{2}-13\)` = **8**

RMSEA = `\(\sqrt{\frac{\frac{\chi^2_T - df_T}{N}}{df_T}}\)` = `\(\sqrt{\frac{\frac{28.978 - 8}{379}}{8}}\)` = **0.083**

CFI = `\(\frac{(\chi^2_0 - df_0)-(\chi^2_T - df_T)}{(\chi^2_0 - df_0)}\)` 
= `\(\frac{(1939.234 - 15)-(28.978 - 8)}{(1939.234 - 15)}\)` = **0.989**

TLI = `\(\frac{(\frac{\chi^2_0} {df_0})-(\frac{\chi^2_T} {df_T})}{(\frac{\chi^2_0} {df_0})-1}\)` = `\(\frac{(\frac{1939.234} {15})-(\frac{28.978} {8})}{(\frac{1939.234} {15})-1}\)` = **0.980**

]

---

# How do we estimate a model? 
## Statistical software (Mplus, R, SAS).

---

<center> Sample CFA Mplus Code </center>

DATA: FILE = mydata.dat;

VARIABLE: 
NAMES = SUP1 SUP2 SUP3;

MODEL: 
SUP by SUP1\* 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; SUP2 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; SUP3; 
SUP@1;

OUTPUT: TECH1;

]

<img class="rectangle" src="images/sup1.png" width="800px"/>
 
]

---

<center> Sample CFA Mplus Estimates </center>

---

<center> Sample CFA Mplus Estimates </center>

MODEL RESULTS

<table class="table" style="font-size: 16px; margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:left;"> </th>
 <th style="text-align:left;"> Estimate </th>
 <th style="text-align:left;"> S.E. </th>
 <th style="text-align:left;"> Est./S.E. </th>
 <th style="text-align:left;"> P-Value </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> SUP BY </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP1 </td>
 <td style="text-align:left;"> 2.202 </td>
 <td style="text-align:left;"> 0.155 </td>
 <td style="text-align:left;"> 14.246 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP2 </td>
 <td style="text-align:left;"> 2.226 </td>
 <td style="text-align:left;"> 0.150 </td>
 <td style="text-align:left;"> 14.879 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP3 </td>
 <td style="text-align:left;"> 1.966 </td>
 <td style="text-align:left;"> 0.164 </td>
 <td style="text-align:left;"> 11.990 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Intercepts </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP1 </td>
 <td style="text-align:left;"> 3.287 </td>
 <td style="text-align:left;"> 0.199 </td>
 <td style="text-align:left;"> 16.522 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP2 </td>
 <td style="text-align:left;"> 2.990 </td>
 <td style="text-align:left;"> 0.196 </td>
 <td style="text-align:left;"> 15.239 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP3 </td>
 <td style="text-align:left;"> 3.322 </td>
 <td style="text-align:left;"> 0.198 </td>
 <td style="text-align:left;"> 16.739 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Variances </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP </td>
 <td style="text-align:left;"> 1.000 </td>
 <td style="text-align:left;"> 0.000 </td>
 <td style="text-align:left;"> 999.000 </td>
 <td style="text-align:left;"> 999.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Resid Var </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP1 </td>
 <td style="text-align:left;"> 0.812 </td>
 <td style="text-align:left;"> 0.183 </td>
 <td style="text-align:left;"> 4.429 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP2 </td>
 <td style="text-align:left;"> 0.543 </td>
 <td style="text-align:left;"> 0.171 </td>
 <td style="text-align:left;"> 3.176 </td>
 <td style="text-align:left;"> 0.001 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP3 </td>
 <td style="text-align:left;"> 1.768 </td>
 <td style="text-align:left;"> 0.243 </td>
 <td style="text-align:left;"> 7.265 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
</tbody>
</table>

]

<img class="rectangle" src="images/sup2.png" width="800px"/>
 
]

---

STDYX Standardization

<table class="table" style="font-size: 16px; margin-left: auto; margin-right: auto;">
 <thead>
 <tr>
 <th style="text-align:left;"> </th>
 <th style="text-align:left;"> Estimate </th>
 <th style="text-align:left;"> S.E. </th>
 <th style="text-align:left;"> Est./S.E. </th>
 <th style="text-align:left;"> P-Value </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> SUP BY </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP1 </td>
 <td style="text-align:left;"> 0.925 </td>
 <td style="text-align:left;"> 0.019 </td>
 <td style="text-align:left;"> 48.366 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP2 </td>
 <td style="text-align:left;"> 0.949 </td>
 <td style="text-align:left;"> 0.017 </td>
 <td style="text-align:left;"> 54.928 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP3 </td>
 <td style="text-align:left;"> 0.828 </td>
 <td style="text-align:left;"> 0.029 </td>
 <td style="text-align:left;"> 28.129 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Intercepts </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP1 </td>
 <td style="text-align:left;"> 1.382 </td>
 <td style="text-align:left;"> 0.117 </td>
 <td style="text-align:left;"> 11.818 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP2 </td>
 <td style="text-align:left;"> 1.275 </td>
 <td style="text-align:left;"> 0.113 </td>
 <td style="text-align:left;"> 11.318 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP3 </td>
 <td style="text-align:left;"> 1.400 </td>
 <td style="text-align:left;"> 0.118 </td>
 <td style="text-align:left;"> 11.897 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Variances </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP </td>
 <td style="text-align:left;"> 1.000 </td>
 <td style="text-align:left;"> 0.000 </td>
 <td style="text-align:left;"> 999.000 </td>
 <td style="text-align:left;"> 999.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Resid Var </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 <td style="text-align:left;"> </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP1 </td>
 <td style="text-align:left;"> 0.143 </td>
 <td style="text-align:left;"> 0.035 </td>
 <td style="text-align:left;"> 4.050 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP2 </td>
 <td style="text-align:left;"> 0.099 </td>
 <td style="text-align:left;"> 0.033 </td>
 <td style="text-align:left;"> 3.011 </td>
 <td style="text-align:left;"> 0.003 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> SUP3 </td>
 <td style="text-align:left;"> 0.314 </td>
 <td style="text-align:left;"> 0.049 </td>
 <td style="text-align:left;"> 6.435 </td>
 <td style="text-align:left;"> 0.000 </td>
 </tr>
</tbody>
</table>

]

<img class="rectangle" src="images/sup3.png" width="800px"/>
 
]

Perceived Social Support (latent SUP) accounts for **85.6%** `\((0.925^2 = 0.856)\)` of the variance in the indicator SUP1. Also, 0.856 + 0.143 = 1.0

---

<center> Sample CFA R (Lavaan) Code </center>

library(lavaan) 
m1 &larr; ' 
SUP =~ NA\*SUP1 + SUP2 + SUP3 
SUP ~~ 1*SUP 
' 
fit1 &larr; cfa(m1, data=mydata, std.lv=T) 
summary(fit1, standardized=T, fit.measures=T, rsquare=T)

]

<img class="rectangle" src="images/sup1.png" width="800px"/>
 
]

---

<center> Sample CFA R (Lavaan) Estimates </center>

---

<center> Sample CFA SAS (Proc Calis) Code </center>

proc calis data=mydata method=ml; 
 path SUP &rarr; SUP1 SUP2 SUP3 = ly1 - ly3; 
 pvar SUP = 1, 
 SUP1 SUP2 SUP3 = te1 - te3; 
run;

]

<img class="rectangle" src="images/sup1.png" width="800px"/>
 
]

---

<center> Sample CFA SAS (Proc Calis) Estimates </center>

<img class="rectangle" src="images/calisresult1.png" width="400px"/>
]

<img class="rectangle" src="images/calisresult2.png" width="400px"/>
 
]

---

# Advantages of CFA? 
## observed vs. latent correlations.

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---

CFA reveals subtle relationships between constructs that may be obscured by scale scores.

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<img class="rectangle" src="images/scalesCFA1.png" width="400px"/>
 
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Disattenuated correlation (controlling for measurement error and specific variance)

---

# How about latent regression? 
## Structural Equation Modeling.

---

CFA reveals subtle relationships between constructs that may be obscured by scale scores.

---

Latent vs. Manifest/Observed

<img class="rectangle" src="images/regression_demo.png" width="400px"/>
]

<img class="rectangle" src="images/SEMest.png" width="400px"/>
 
]

Measurement error in the observed variables can reduce the accuracy of the regression model.

Model Fit: χ2(24, n=144) = 36.14; RMSEA = .059(.000;.097) ; CFI = .980; TLI/NNFI = .970

---

# Does the relationship between X and Y depend on Z? 
## Moderation.

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]

]

Mplus code using the Latent Moderated Structural Equations (LMS) Method.

---

<center><medium> Regular Moderation </medium></center>

`\(R^2 = 0.11\)`

]

<center><medium> Latent Moderation </medium></center>

`\(R^2 = 0.46\)`

]

**Model differences**: In *regular moderation*, measurement error can inflate predictive effects (i.e., making them seem larger). *Latent moderation* provides more accurate estimates.

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# Does the effect of X on Y operate through M? 
## Mediation.

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]

<center><medium> Latent Mediation </medium></center>

]

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# Do I have enough power for this? 
## SEM Power Analysis.

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Parameter Power

]

<img class="rectangle" src="images/montecarlo3.png" width="800px"/>
 
]

80% power for a correlation of at least -0.30 given *N* = 150.

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Model Power

]

<img class="rectangle" src="images/rmseapower2.png" width="800px"/>
 
]

80% power for a CFA with at least 60 df given *N* = 85.

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#Thanks!
 
## Any further questions?

## Feel free to join the coding session.

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