What analysis to use for one quantitative independent variable and multiple quantitative dependent variables?

Question

My data is based on survey responses. The independent variable is aggregated based on three Likert questions between rated $1$ to $5$ (IV sums these three and divides by $3$). The four dependent variables are Likert questions between $1$ to $5$.

The obvious (but perhaps controversial) solution is to run four simple linear regression models, one per dependent variable. However, my concern is finding significant results from chance alone.

I looked into multivariate multiple linear regression, but this analysis requires more than one IV, which does not help my case.

Answer 1

Your independent variable appears to be a composite of three items which is supposed to represent something intangible (e.g. anxiety, socioeconomic status) and four DVs which may also represent some intangible. Specifically, the items are manifest variables because they are directly observable and the intangibles or the constructs you are trying to capture are latent variables because they are indirectly observed variables via the manifest variables.

I feel the most obvious solution to your problem is some kind of structural equation model (SEM), where the IV latent variable is regressed on the DV latent variable. Here is an overly simplified simulation of your data (some of what I write here is a bit lazy but still demonstrates what I am trying to convey). I simulate the data in R and fit the model with the lavaan package, which is a common package for SEM.

#### Simulated Latents ####
set.seed(123)
n <- 1000  # number of observations
IV <- rnorm(n)  # latent variable IV
DV <- 0.5*IV + rnorm(n)  # latent variable DV

#### Simulated Manifests ####
IV1 <- IV + rnorm(n)
IV2 <- IV + rnorm(n)
IV3 <- IV + rnorm(n)
DV1 <- DV + rnorm(n)
DV2 <- DV + rnorm(n)
DV3 <- DV + rnorm(n)
DV4 <- DV + rnorm(n)

#### Combine Data ####
dat <- data.frame(IV1, IV2, IV3, DV1, DV2, DV3, DV4)

#### Construct SEM Model ####
model <- '
  # latent variable definitions
  IV =~ IV1 + IV2 + IV3
  DV =~ DV1 + DV2 + DV3 + DV4

  # regression
  DV ~ IV
'

#### Fit Model ####
library(lavaan)
fit <- sem(model, data = dat)
summary(fit, fit.measures = TRUE)

#### Plot ####
semPlot::semPaths(fit)

The plot below shows the constructed model (unlabeled):

The circles represent the latent variables (IV and DV) and the squares are the items that represent them, or the manifest variables. The lines drawn between the circles and squares are regression paths which estimate how much each item "loads" onto the latent variable, or essentially how close it relates to the latent variable. The semi-circle arrow paths are the variances for each.

That information isn't easy to see, so I change the code here. The standardized loadings are shown below, where each number on the arrows represents how much each item "loads" onto the latent variable. You can see that the arrow between the IV and DV represents the regression path, which shows that the relationship between the two is $\beta = .51$, which is very close to what we specified in our simulated data:

#### Plot ####
semPlot::semPaths(
  fit,
  "std",
  layout = "spring",
  label.cex=1,
  edge.label.cex = 1.5
  )

The full model summary can be run with summary(fit, fit.measures = T), which I do not go into detail here, but examining them is a necessary part of fitting these models:

lavaan 0.6.16 ended normally after 29 iterations

  Estimator                                         ML
  Optimization method                           NLMINB
  Number of model parameters                        15

  Number of observations                          1000

Model Test User Model:
                                                      
  Test statistic                                23.797
  Degrees of freedom                                13
  P-value (Chi-square)                           0.033

Model Test Baseline Model:

  Test statistic                              2476.670
  Degrees of freedom                                21
  P-value                                        0.000

User Model versus Baseline Model:

  Comparative Fit Index (CFI)                    0.996
  Tucker-Lewis Index (TLI)                       0.993

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)             -11432.560
  Loglikelihood unrestricted model (H1)     -11420.661
                                                      
  Akaike (AIC)                               22895.119
  Bayesian (BIC)                             22968.736
  Sample-size adjusted Bayesian (SABIC)      22921.095

Root Mean Square Error of Approximation:

  RMSEA                                          0.029
  90 Percent confidence interval - lower         0.008
  90 Percent confidence interval - upper         0.047
  P-value H_0: RMSEA <= 0.050                    0.975
  P-value H_0: RMSEA >= 0.080                    0.000

Standardized Root Mean Square Residual:

  SRMR                                           0.020

Parameter Estimates:

  Standard errors                             Standard
  Information                                 Expected
  Information saturated (h1) model          Structured

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)
  IV =~                                               
    IV1               1.000                           
    IV2               1.014    0.059   17.161    0.000
    IV3               1.031    0.060   17.130    0.000
  DV =~                                               
    DV1               1.000                           
    DV2               1.097    0.049   22.448    0.000
    DV3               1.055    0.049   21.604    0.000
    DV4               1.093    0.050   21.909    0.000

Regressions:
                   Estimate  Std.Err  z-value  P(>|z|)
  DV ~                                                
    IV                0.565    0.048   11.664    0.000

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)
   .IV1               0.929    0.062   14.940    0.000
   .IV2               0.961    0.064   14.978    0.000
   .IV3               1.014    0.067   15.143    0.000
   .DV1               1.007    0.058   17.476    0.000
   .DV2               0.913    0.058   15.833    0.000
   .DV3               1.051    0.061   17.143    0.000
   .DV4               1.048    0.063   16.729    0.000
    IV                0.972    0.087   11.115    0.000
   .DV                0.886    0.075   11.787    0.000

This is just scratching the surface but gives you at least a conceptual introduction to what you can do for your case. To learn more, a good starting place is either Kline's book for conceptual knowledge or Beaujean's book for programming it in R.

Answer 2

Useful comments/answer were already given to make you think if you really should do what you planned to do. Still, if you would nevertheless like to continue, I'll briefly mention below how that can be done...

You could arrange your data in a long format, 4 records for each person, one for each dependent variable, containing the value of the given dependent and the value of the independent, which is the same for all four records. Also, add a number 1-4, specifying which dependent variable the record refers to!

With these data you could run a "multivariate response model" like e.g. described here in chapter 14. In spss you can run such a model using procedure "mixed" in combination with the "repeated" option and an unstructured covariance matrix. In R you can do this using e.g. package glmmTMB in combination with option "dispformula=~0" and also the unstructured covariance matrix; also glm from package nlme in R can be used, again with unstructured covariance matrix.

The unstructured covariance matrix is important here because you would probably want a different variance for each of your dependents, and different covariances or correlations as well. More simple structures can be used too...