This vignette proposes a novel approach to decompose and identify variance components in any statistical parameter obtained from a specification curve analysis. It can help to understand which analytical choices explain most of the variance in the specification curve (e.g., the variance in the effect sizes).

In order to have some data to work with, we again run the minimal example included in the package.

We can think of the specification curve analysis as a factorial design in which we investigate the influence of different types of analytical choices on a resulting coefficient or test statistic. One way to investigate these effects is to ask how much variance in the specification curve is explained by which analytical choice (or combination of analytical choices).

In a first step, we estimate a simple multilevel model without predictors (a so-called null model) with specific random effects that represent the analytical choices. We can used the function `lmer()`

from the package lme4 for the estimation.

```
# Package to estimate multilevel models
library(lme4)
# Estimate model
m1 <- lmer(estimate ~ 1 + (1|x) + (1|y) + (1|controls) + (1|subsets), data = results)
# Check model summary
summary(m1)
```

```
## Linear mixed model fit by REML ['lmerMod']
## Formula: estimate ~ 1 + (1 | x) + (1 | y) + (1 | controls) + (1 | subsets)
## Data: results
##
## REML criterion at convergence: 591.7
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.85509 -0.60698 -0.03593 0.55041 2.83207
##
## Random effects:
## Groups Name Variance Std.Dev.
## subsets (Intercept) 0.76138 0.8726
## controls (Intercept) 0.03861 0.1965
## y (Intercept) 19.72906 4.4417
## x (Intercept) 0.42335 0.6507
## Residual 1.00930 1.0046
## Number of obs: 192, groups: subsets, 12; controls, 4; y, 2; x, 2
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 4.079 3.187 1.28
```

We have stored the random effects in the object `m1`

. We already see that most variance is related to the random effect `y|intercept`

. Yet, how much exactly?

Specr offers two ways to decompose the variance of the specification curve. First, we can simply compute the intraclass correlation coefficients using the function `icc_specs()`

.

```
## grp vcov icc percent
## 1 subsets 0.76 0.03 3.47
## 2 controls 0.04 0.00 0.18
## 3 y 19.73 0.90 89.83
## 4 x 0.42 0.02 1.93
## 5 Residual 1.01 0.05 4.60
```

We can see that 89.8% of the variance in the obtained results is related to choosing different dependent variables. Only small parts of the variance are explained by other analytical choices.

Second, we can alternatively use the function `plot_variance()`

to obtain a visualization. The function calls `icc_specs()`

automatically. We can hence pass the multilevel results object directly. Further customization via ggplot2 is possible.

You might ask yourself why we did not include the multilevel model estimation in the `icc_specs()`

function. The reason is straight-forward: In many cases, it might be of interest to decompose the variance in different ways (e.g., including interactions between the analytical choices). For example, we could specify the a new multilevel model that includes alternative random effects (e.g., interactions between the analytical choices) and we can again obtain a result table or a plot.

```
m2 <- lmer(estimate ~ 1 + (1|x) + (1|y) + (1|controls) + (1|subsets) + (1|x:y) + (1|y:controls) + (1|y:subsets), data = results)
# Get table
icc_specs(m2) %>%
mutate_if(is.numeric, round, 2)
```

```
## grp vcov icc percent
## 1 y:subsets 0.76 0.03 3.49
## 2 subsets 0.41 0.02 1.87
## 3 y:controls 0.00 0.00 0.00
## 4 x:y 0.02 0.00 0.09
## 5 controls 0.05 0.00 0.22
## 6 y 19.46 0.90 89.55
## 7 x 0.42 0.02 1.92
## 8 Residual 0.62 0.03 2.87
```

We can see that the interaction between analytical choices relating to subsets and dependent variables explains 3.5%.

Finally it should be noted that any other parameter in the data frame can be investigated as well (e.g., not only the estimate such as a regression coefficient, but also the p.value).

`## boundary (singular) fit: see ?isSingular`

```
## grp vcov icc percent
## 1 subsets 0.00 0.04 4.07
## 2 controls 0.00 0.00 0.00
## 3 y 0.02 0.30 29.99
## 4 x 0.00 0.01 0.74
## 5 Residual 0.03 0.65 65.21
```