# Frequentist Regression in Mplus

## Introduction

This tutorial provides the reader with a basic tutorial how to perform a regression analysis in Mplus. Throughout this tutorial, the reader will be guided through importing datafiles, exploring summary statistics and regression analyses. Here, we will exclusively focus on frequentist statistics.

We are continuously improving the tutorials so let me know if you discover mistakes, or if you have additional resources I can refer to. The source code is available via Github. If you want to be the first to be informed about updates, follow me on Twitter.

## Preparation

This tutorial expects:

• any version of Mplus.This tutorial was made using Mplus version 8_3.
• Basic knowledge of hypothesis testing
• Basic knowledge of correlation and regression
• Basic knowledge of coding in Mplus

## Example Data

The data we will be using for this exercise is based on a study about predicting PhD-delays (Van de Schoot, Yerkes, Mouw and Sonneveld 2013).The data can be downloaded here. Among many other questions, the researchers asked the Ph.D. recipients how long it took them to finish their Ph.D. thesis (n=333). It appeared that Ph.D. recipients took an average of 59.8 months (five years and four months) to complete their Ph.D. trajectory. The variable B3_difference_extra measures the difference between planned and actual project time in months (mean=9.97, minimum=-31, maximum=91, sd=14.43). For more information on the sample, instruments, methodology and research context we refer the interested reader to the paper.

For the current exercise we are interested in the question whether age (M = 31.7, SD = 6.86) of the Ph.D. recipients is related to a delay in their project.

The relation between completion time and age is expected to be non-linear. This might be due to that at a certain point in your life (i.e., mid thirties), family life takes up more of your time than when you are in your twenties or when you are older.

So, in our model the $$gap$$ (B3_difference_extra) is the dependent variable and $$age$$ (E22_Age) and $$age^2$$(E22_Age_Squared ) are the predictors. The data can be found in the file phd-delays_nonames.csv . (In Mplus the first row CANNOT have the variable names, these have already been deleted for you)

##### Question: Write down the null and alternative hypotheses that represent this question. Which hypothesis do you deem more likely?

$$H_0:$$ $$age$$ is not related to a delay in the PhD projects.

$$H_1:$$ $$age$$ is related to a delay in the PhD projects.

$$H_0:$$ $$age^2$$ is not related to a delay in the PhD projects.

$$H_1:$$ $$age^2$$is related to a delay in the PhD projects.

## Preparation – Importing and Exploring Data

You can find the data in the file phd-delays_nonames.csv , which contains all variables that you need for this analysis. Although it is a .csv-file, you can directly load it into Mplus using the following syntax:

TITLE: Mplus analysis summary
DATA: FILE IS phd-delays_nonames.csv;
VARIABLE: NAMES ARE diff child sex Age Age2;
USEVARIABLES ARE diff Age Age2;
OUTPUT: sampstat;

Once you loaded in your data, it is advisable to check whether your data import worked well. Therefore, first have a look at the summary statistics of your data. You can do this by looking at the sampstat ouput.

##### Question: Have all your data been loaded in correctly? That is, do all data points substantively make sense? If you are unsure, go back to the .csv-file to inspect the raw data.
MODEL RESULTS

Two-Tailed
Estimate       S.E.  Est./S.E.    P-Value

Means
DIFF               9.967      0.790     12.622      0.000
AGE               31.676      0.375     84.433      0.000
AGE2            1050.217     35.916     29.241      0.000

The descriptive statistics make sense:

$$diff$$: Mean (9.97), SE (0.79)

$$Age$$: Mean (31.68), SE (0.38)

$$Age^2$$: Mean (1050.22), SE (35.92)

## Plot

Before we continue with analyzing the data we can also plot the expected relationship. We can do this by adding the following code to the input syntax of Mplus and then in the top menu go to Plot > View plot > Scatterplots (sample value) > View > Set X to AGE > OK

PLOT:
TYPE IS PLOT1; 

This gives us the following plot: There seems to be some quadratic relationship

## Regression Analysis

Now, let’s run a multiple regression model predicting the difference between Ph.D. students’ planned and actual project time by their age (note that we ignore assumption checking, if you want a quick introduction to the assumptions underlying a regression, please have look at https://statistics.laerd.com/spss-tutorials/linear-regression-using-spss-statistics.php).

To run a multiple regression with Mplus, you first specify the model, then fit the model and finally acquire the summary. The model is specified as follows:

1. We set a title of the file under the TITLE: command
2. We use a DATA command and we tell Mplus what the datafile is called.
3. In the next syntax line, we use a VARIABLE command that consists of two lines. First, we tell Mplus what the variable names are by using the NAMES ARE statement. Note that the order of the variable names has to mirror the actual order in the dataset. Second, we tell Mplus which variables we are actually going to use by using USEVARIABLES ARE. This way Mplus knows which columns in the .csv file to use.
4. We specified a MODEL where an outcome variable (diff) is being regressed ON two predictors (Age and Age2). Dependent or Y variables always appear on the left hand side of the ON statement and independent or X variables always appear on the right hand side of the ON statement.
5. We specify what types of output we would like after the OUTPUT statement. See here for a summary of all possible outputs and syntax

Now, preform a multiple linear regression and answer the following questions:

##### Question: Using a significance criterion of 0.05, is there a significant effect of $$age$$ and $$age^2$$?
TITLE: Frequentist analysis

DATA: FILE IS phd-delays_nonames.csv;

VARIABLE: NAMES ARE diff child sex Age Age2; ! All the variables in the dataset

USEVARIABLES ARE diff Age Age2; ! The variables we use in this analysis

MODEL:
diff ON Age (Beta_Age);   ! Regression coefficient 1.
diff ON Age2(Beta_Age2);  ! Regression coefficient 2

OUTPUT: sampstat;
MODEL RESULTS

Two-Tailed
Estimate       S.E.  Est./S.E.    P-Value

DIFF     ON
AGE                2.657      0.583      4.554      0.000
AGE2              -0.026      0.006     -4.236      0.000

Intercepts
DIFF             -47.088     12.285     -3.833      0.000

Residual Variances
DIFF             194.641     15.084     12.903      0.000

There is a significant effect of $$age$$ and $$age^2$$, with b=2.657, p <.001 for $$age$$, and b=-0.026, p<.001 for $$age^2$$.

Surveys in academia have shown that a large number of researchers interpret the p-value wrong and misinterpretations are way more widespread than thought. Have a look at the article by Greenland et al. (2016) that provides a guide to clear and concise interpretations of p.

##### Question: What can you conclude about the hypothesis being tested using the correct interpretation of the p-value?

Assuming that the null hypothesis is true in the population, the probability of obtaining a test statistic that is as extreme or more extreme as the one we observe is <0.1%. Because the effect of $$age^2$$ is below our pre-determined alpha level, we reject the null hypothesis.

Recently, a group of 72 notable statisticians proposed to shift the significance threshold to 0.005 (Benjamin et al. 2017, but see also a critique byTrafimow, …, Van de Schoot, et al., 2018). They argue that a p-value just below 0.05 does not provide sufficient evidence for statistical inference.

Because the p-values for both regression coefficients were really small <.001, the conclusion doesn’t change in this case.

Of course, we should never base our decisions on single criterions only. Luckily, there are several additional measures that we can take into account. A very popular measure is the confidence interval. To get the confidence intervals for the model parameters, you can add CINTERVAL to the OUTPUT statement in the input file, like

OUTPUT: sampstat CINTERVAL;

This will give this output:

CONFIDENCE INTERVALS OF MODEL RESULTS

Lower .5%  Lower 2.5%    Lower 5%    Estimate    Upper 5%  Upper 2.5%   Upper .5%

DIFF     ON
AGE              1.154       1.514       1.697       2.657       3.617       3.801       4.160
AGE2            -0.042      -0.038      -0.036      -0.026      -0.016      -0.014      -0.010

Intercepts
DIFF           -78.732     -71.167     -67.297     -47.088     -26.880     -23.010     -15.445

Residual Variances
DIFF           155.787     165.076     169.827     194.641     219.455     224.207     233.496
##### Question: What can you conclude about the hypothesis being tested using the correct interpretation of the confidence interval?

$$Age$$: 95% CI [1.514, 3.801]

$$Age^2$$: 95% CI [-0.038, -0.014]

In both cases the 95% CI’s don’t contain 0, which means, the null hypotheses should be rejected. A 95% CI means, that, if infinitely samples were taken from the population, then 95% of the samples contain the true population value. But we do not know whether our current sample is part of this collection, so we only have an aggregated assurance that in the long run if our analysis would be repeated our sample CI contains the true population parameter.

Additionally, to make statements about the actual relevance of your results, focusing on effect size measures is inevitable.

##### Question: What can you say about the relevance of your results? Focus on the explained variance and the standardized regression coefficients.

To get the standardized results and an estimate of the explained variance (R$$^2$$), you can add CINTERVAL to the OUTPUT statement in the input file, like

OUTPUT: sampstat CINTERVAL STDYX;
R-SQUARE

Observed                                        Two-Tailed
Variable        Estimate       S.E.  Est./S.E.    P-Value

DIFF               0.063      0.026      2.435      0.015

R$$^2$$= 0.063 in the regression model. This means that 6.3% of the variance in the PhD delays, can be explained by $$age$$ and $$age^2$$.

STDYX Standardization

Two-Tailed
Estimate       S.E.  Est./S.E.    P-Value

DIFF     ON
AGE                1.262      0.269      4.685      0.000
AGE2              -1.174      0.271     -4.340      0.000

Intercepts
DIFF              -3.268      0.819     -3.992      0.000

Residual Variances
DIFF               0.937      0.026     36.472      0.000

The standardized coefficients, age (1.262) and age$$^2$$ (-1.174), show that the effects of both regression coefficients are comparable, but the effect of age is somewhat higher. This means that the linear effect of age on PhD delay (age) is a bit larger than the quadratic effect of age on PhD delay (age2)

Only a combination of different measures assessing different aspects of your results can provide a comprehensive answer to your research question.

##### Question: Drawing on all the measures we discussed above, formulate an answer to your research question.

The variables $$age$$ and $$age^2$$ are significantly related to PhD delays. However, the total explained variance by those two predictors is only 6.3%. Therefore, a large part of the variance is still unexplained.