I'm struggling with a linear model.

My dependent variable is change in central anterior chamber depth of the eye with time. Measurement 1 is time 0, measurement 2 is 6 years later and I am regressing the change with predictor variables.

My dependent variable is not normally distributed (see histogram).

I have developed 5 hierarchical models each with less significant terms added on.

All of the models show heteroskedasticity and non-normal errors.

I want to report associations but don't want to get things wrong, so any help would be appreciated.

So, model '210' is the simplest model where the anterior chamber depth change is most correlated with the starting depth (I think this association is too strong to be explained by 'regression to the mean' but opinions would be appreciated).

Code:

```
> model210<-lm(R_ACD_change~Racd_screen_median,ACDdata[(ACDdata$Pi_Bl1==0 & ACDdata$Racd_screen_median<3),])
> summary(model210)
Call:
lm(formula = R_ACD_change ~ Racd_screen_median, data = ACDdata[(ACDdata$Pi_Bl1 ==
0 & ACDdata$Racd_screen_median < 3), ])
Residuals:
Min 1Q Median 3Q Max
-0.39377 -0.12872 -0.02327 0.07721 0.74808
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.2766 0.1787 7.145 2.13e-11 ***
Racd_screen_median -0.5908 0.0737 -8.015 1.31e-13 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1868 on 181 degrees of freedom
Multiple R-squared: 0.262, Adjusted R-squared: 0.2579
F-statistic: 64.25 on 1 and 181 DF, p-value: 1.312e-13
```

The next model adds the angle width as a predictor

Code:

```
> model220<-lm(R_ACD_change~Racd_screen_median+R_median_shaff,ACDdata[(ACDdata$Pi_Bl1==0 & ACDdata$Racd_screen_median<3),])
> summary(model220)
Call:
lm(formula = R_ACD_change ~ Racd_screen_median + R_median_shaff,
data = ACDdata[(ACDdata$Pi_Bl1 == 0 & ACDdata$Racd_screen_median <
3), ])
Residuals:
Min 1Q Median 3Q Max
-0.37483 -0.12239 -0.02013 0.08611 0.61278
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.21075 0.16842 7.189 1.71e-11 ***
Racd_screen_median -0.64229 0.07000 -9.175 < 2e-16 ***
R_median_shaff 0.09470 0.01852 5.115 8.03e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1755 on 179 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.3557, Adjusted R-squared: 0.3485
F-statistic: 49.4 on 2 and 179 DF, p-value: < 2.2e-16
```

The next model adds the follow-up lens thickness/eye length ratio. This was only available at follow-up so is not a 'predictor' but is associated with the change.

Code:

```
> model230<-lm(R_ACD_change~Racd_screen_median+R_median_shaff+R_Lens_AL,ACDdata[(ACDdata$Pi_Bl1==0 & ACDdata$Racd_screen_median<3),])
> summary(model230)
Call:
lm(formula = R_ACD_change ~ Racd_screen_median + R_median_shaff +
R_Lens_AL, data = ACDdata[(ACDdata$Pi_Bl1 == 0 & ACDdata$Racd_screen_median <
3), ])
Residuals:
Min 1Q Median 3Q Max
-0.35673 -0.09986 -0.01836 0.09092 0.54430
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.34071 0.25461 9.193 < 2e-16 ***
Racd_screen_median -0.66705 0.06598 -10.110 < 2e-16 ***
R_median_shaff 0.07861 0.01765 4.454 1.56e-05 ***
R_Lens_AL -4.87701 0.86730 -5.623 7.91e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.16 on 164 degrees of freedom
(15 observations deleted due to missingness)
Multiple R-squared: 0.4631, Adjusted R-squared: 0.4533
F-statistic: 47.16 on 3 and 164 DF, p-value: < 2.2e-16
```

The final model has the change in angle width as well

Code:

```
> summary(model240)
Call:
lm(formula = R_ACD_change ~ Racd_screen_median + R_Shaffer_Change +
R_median_shaff + R_Lens_AL, data = ACDdata[(ACDdata$Pi_Bl1 ==
0 & ACDdata$Racd_screen_median < 3), ])
Residuals:
Min 1Q Median 3Q Max
-0.32231 -0.10900 -0.01588 0.08898 0.48960
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.21582 0.24535 9.031 5.01e-16 ***
Racd_screen_median -0.68331 0.06315 -10.821 < 2e-16 ***
R_Shaffer_Change 0.04892 0.01444 3.387 0.000889 ***
R_median_shaff 0.09389 0.01789 5.247 4.81e-07 ***
R_Lens_AL -4.04595 0.84703 -4.777 3.98e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1519 on 161 degrees of freedom
(17 observations deleted due to missingness)
Multiple R-squared: 0.5038, Adjusted R-squared: 0.4915
F-statistic: 40.87 on 4 and 161 DF, p-value: < 2.2e-16
```

I'm mainly interested in reporting the association with starting angle depth (which seems strong) and with angle width (which is weak but interesting). I'm also quite interested in reporting the association with follow-up lens thickness as lens thickness changes are thought to contribute to this.

Does this seem reasonable or do I need to consider robust regression instead.

Very very many thanks for looking at this.