Summary

Increase in community involvement is significantly associated with a significant decrease in loneliness, p < .0001. The change in community involvement between ages 17 and 19 is not a significant predictor of loneliness scores, p > .2. However, it is significant between the ages of 19 and 23, along with 17 and 23, suggesting a non-linear effect over time. Orthogonal polynomial contrasts indicate that there is a quadratic relationship over time, p = .0053.

There is a significant increase in self-esteem scores between ages 17 and 19, p = .0049. There is also a significant increase in self-esteem scores between ages 17 and 23, p < .0001. An increase in community involvement is significantly associated with an increase in self-esteem scores, p = .0021.

There is a significant decrease in depression scores between ages 17 and 19, p = .0057. There is also a significant decrease in depression scores between ages 17 and 23, p < .0001. However, there is not a significant association between these decreases and community involvement, p > .2.

Predicting loneliness from community involvement

To visualize the relationship between community involvement and loneliness across ages, it is easiest to look at change in the two continuous variables separately in terms of age.

As you can see from the first set of graphs, there is an obvious drop-off in loneliness between ages 19 and 23; which lines from each participant show is extremely consistent within these participants (the lines are colored purely to see individual participants better since there are so many). The change is much more pronounced between these two ages than between 17 and 19; suggesting a non-linear effect.



A standard multilevel model (not shown) using the reference point of age 17 confirms that there is a not a significant effect of community involvement on loneliness scores between ages 17 and 19, but a significant effect between ages 17 and 23. To examine the non-linear effect and provide a more accurate model, we use orthogonal polynomial contrast matrix:

\[ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ Age \\ \ \ \ \ \ \ \ \ \ \ \ \ \textbf{17}\ \ \ \textbf{19} \ \ \ \textbf{23} \\ \begin{matrix} Lin \\ Quad \end{matrix} \begin{bmatrix} -.5 & .5 & 0 \\ .5 & .5 & -1 \\ \end{bmatrix} \] Now look at the fixed effects.

##                     Estimate Std. Error       df    t value       Pr(>|t|)
## (Intercept)      33.71236192 0.60988124 713.5340 55.2769293 0.000000000000
## lin               1.04336893 1.25315765 425.7569  0.8325919 0.405541698710
## quad             20.81002871 0.98235023 578.7607 21.1839201 0.000000000000
## cominvScore       0.07977823 0.01790929 678.4603  4.4545728 0.000009829874
## lin:cominvScore   0.02945904 0.03565478 409.1056  0.8262299 0.409155295659
## quad:cominvScore  0.08237311 0.02939903 523.1051  2.8018989 0.005268543653

The first part of the output shows the interaction of age and community involvement. As stated above, the linear effect comparing ages 17 and 19 is not significant (p = .299), while the quadratic effect comparing ages 17 and 19 to age 23 is (p<.001). I should note that the coefficient is positive, but it is actually representing a decrease because the highest age was coded as a negative value.


Second, we have the variances, standard deviations, and correlations of our random effects.

##  Groups   Name        Variance Std.Dev. Corr       
##  id       (Intercept)  33.769   5.8111             
##           lin          36.206   6.0172  0.047      
##           quad        102.841  10.1410  0.692 0.344
##  Residual              42.810   6.5429

The most important part of this second output is that the quadratic effect accounts for a high difference in ages above individual differences and the linear effect.

Predicting self-esteem from community involvement

The only thing we can see in self-esteem graphically is that it tends to increase over time. Since this effect seems present for both ages, we have no graphical reason to assume that there is an interaction or a non-linear effect. Therefore, we look at the standard model that examines change in scores over time.

##                Estimate Std. Error        df   t value     Pr(>|t|)
## (Intercept) 66.12515725 0.83030338 1393.1868 79.639754 0.000000e+00
## Age19        1.58853380 0.56264311  387.7562  2.823342 4.997674e-03
## Age23        5.91767348 0.73846869  380.7633  8.013439 1.376677e-14
## cominvScore  0.06325529 0.02055266 1444.6737  3.077717 2.125165e-03

So we can confirm that there is a signicant increase between self-esteem scores at ages 17 and 19 (p = .0049), along with 17 and 23 (p < .0001). Most importantly, an increase in community involvement across ages is significantly associated with an increase in self-esteem (p = .0021).

##  Groups   Name        Variance Std.Dev. Corr         
##  id       (Intercept) 172.558  13.1361               
##           Age19        24.376   4.9372  -0.275       
##           Age23       104.069  10.2014  -0.569  0.619
##  Residual              47.027   6.8576

And here are the variances, standard deviations, and correlations of the random effects.

Predicting depression from community involvement

##                 Estimate Std. Error        df    t value    Pr(>|t|)
## (Intercept) 17.716241912 0.58875318 1368.6105 30.0911190 0.000000000
## Age19       -1.240211731 0.44648595  406.6167 -2.7777172 0.005727675
## Age23       -4.767925821 0.51678896  402.3097 -9.2260597 0.000000000
## cominvScore -0.001286368 0.01466226 1471.5002 -0.0877333 0.930100586

The fixed effects indicate that there is a significant decrease in depression scores between ages 17 and 19 (p = .0057), along with 17 and 23 (p < .0001). However, community involvement has an extremely high p-value and is non-significant.

##  Groups   Name        Variance Std.Dev. Corr         
##  id       (Intercept) 72.173   8.4955                
##           Age19       12.051   3.4715   -0.389       
##           Age23       35.769   5.9807   -0.626  0.960
##  Residual             32.947   5.7399

And here are the variances, standard deviations, and correlations of the random effects.

Equation using orthogonal polynomials (UNFINISHED SECTION)

Unsimplified equation:

\[ a \] This can be simplified into the equation \[ Y_{hi}= \sum_{p=0}^P(\alpha_{p0} + \alpha_{p1}\overline{X}_{i} + \mu_{pi})c_{ph} + \epsilon_{hi} \] The subscripts are interpreted as follows:

h = Measurement occasion (Age)
i = Participant
p = Polynomial order

The coefficients can be interpreted as follows:

c = contrast of the pth order polynomial
αp0 =
αp1 = the mean difference between age groups in respect to polynomial effect of order p