# Chapter 14 Area under Curve, AULCSF and \(R^2\) of the CSF

### Citing the Guide and the Package

If you have used the **smCSF** package for your research, please cite the paper below:

**Min, S. H.**, & Reynaud, A. (2024). Applying Resampling and Visualization Methods in Factor Analysis to Model Human Spatial Vision., 65(1), 17-17.*Investigative Ophthalmology & Visual Science*

### Getting Started

As in the previous chapter, we will use data of achromatic contrast sensitivity from 51 normal observers from this paper (a subset of the entire dataset) and the **smCSF** package:

Kim, Y. J., Reynaud, A., Hess, R. F., & Mullen, K. T. (2017). A normative data set for the clinical assessment of achromatic and chromatic contrast sensitivity using a qCSF approach. ** Investigative Ophthalmology & Visual Science**, 58(9), 3628-3636.

Let’s load the necessary packages.

```
## # A tibble: 6 × 4
## Subject Repetition SpatialFreq Sensitivity
## <chr> <dbl> <dbl> <dbl>
## 1 S1 1 0.25 23.8
## 2 S2 1 0.25 18.0
## 3 S3 1 0.25 14.2
## 4 S4 1 0.25 5.14
## 5 S5 1 0.25 16.0
## 6 S6 1 0.25 10.9
```

There are four columns in this data frame:

First,

`Subject`

refers to each participant. There are 51 participants total in the .csv file.Next,

`Repetition`

refers to each repetition of the measurement The participants performed two repetitions of the contrast sensitivity measurement.`SpatialFreq`

refers to spatial frequency of the stimuli that were shown to test the observers’ contrast sensitivity. There are a total of 12 spatial frequencies.The

`Sensitivity`

column refers to the linear values of the contrast sensitivity.

## 14.1 Area under a Curve

Area under a curve under the contrast sensitivity function (CSF) as an index of visual performance or sensitivity. The higher the areal measure, the better the visual performance. There are two ways to calculate the area: 1) trapezoidal integration of the contrast sensitivity data as a function of spatial frequency, and 2) area under the fitted log CSF (AULCSF).

Let’s create a data frame `ACh_avg`

that contains the averaged contrast sensitivity data for each repetition and spatial frequency across 51 subjects.

```
ACh_avg <- ACh %>%
group_by(Repetition, SpatialFreq) %>%
summarise(
avgSens = mean(Sensitivity),
stdErrSens = sm_stdErr(Sensitivity),
stdDevSens = sd(Sensitivity)
)
ACh_avg
```

```
## # A tibble: 24 × 5
## # Groups: Repetition [2]
## Repetition SpatialFreq avgSens stdErrSens stdDevSens
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.25 12.6 0.743 5.31
## 2 1 0.35 14.7 0.777 5.55
## 3 1 0.49 18.6 0.993 7.09
## 4 1 0.68 23.7 1.22 8.71
## 5 1 0.94 28.7 1.36 9.68
## 6 1 1.31 32.0 1.40 9.99
## 7 1 1.83 32.4 1.43 10.2
## 8 1 2.54 29.6 1.39 9.93
## 9 1 3.54 24.3 1.22 8.70
## 10 1 4.93 18.1 0.960 6.85
## # ℹ 14 more rows
```

As in the previous chapter, lets store the data into proper data frames for each repetition.

```
ACh_avg$Repetition <- factor(ACh_avg$Repetition) # factor
ACh_avg1 <- ACh_avg %>% filter(Repetition == 1) # repetition 1
ACh_avg2 <- ACh_avg %>% filter(Repetition == 2) # repetition 2
```

Here is an example of trapezoidal integration of the contrast sensitivity function (i.e., AUC). AUC is an abbreviation of **a**rea **u**nder a **c**urve from trapezoidal integration.

```
ACh_avg2 %>% ggplot(aes(x = SpatialFreq, y = avgSens)) +
geom_area(fill = sm_color("lightgreen"), alpha = 0.5) +
geom_point(color = sm_color("viridian")) +
geom_line(color = sm_color("viridian")) +
sm_hgrid() +
xlab("Spatial frequency (c/deg)") +
ylab("Averaged sensitivity") +
ggtitle("Repetition 2 (n=51)") +
scale_x_continuous(trans = "log10") +
scale_y_continuous(trans = "log10") +
scale_y_log10(limits = c(1, 100)) +
annotate("text", x = 0.6, y = 1.3, label = "Trapezoidal AUC")
```

Here is an example of the AULCSF.

```
ACh_avg2 %>% ggplot(aes(x = SpatialFreq, y = avgSens)) +
sm_areaCSF(fill = sm_color("lightgreen"), alpha = 0.5) +
sm_CSF(color = sm_color("viridian"), size = .8) +
geom_point(color = sm_color("viridian")) +
sm_hgrid() +
xlab("Spatial frequency (c/deg)") +
ylab("Averaged sensitivity") +
ggtitle("Repetition 2 (n=51)") +
scale_y_log10(limits = c(1, 100)) +
annotate("text", x = 0.4, y = 1.3, label = "AULCSF")
```

Notice that these that AUC and AULCSF are slightly different. AUC denotes the area that is enclosed by the lines that connect each point. AULCSF refers to the area that is enclosed by the fitted contrast sensitivity function. The astute reader may realize that if the fit of the contrast sensitivity function is poor, then the AULCSF might not be accurate. Hence, it might be necessary to prove that the CSF fit is robust with the given data. To avoid these hassles, some researchers use AUC instead. Nevertheless, **smCSF** offers functions that compute both AUC and AULCSF.

The higher the AUC and AULCSF are, the higher the overall sensitivity of the observer. Therefore, when potential treatment is being studied, it is of interest to see an increase in AUC and AULCSF in the visually impaired after their treatment.

## 14.2 `sm_trapz()`

, `sm_AULCSF()`

and `sm_r2()`

This section is identical to what we have seen in Chapter 6. For those who have read Chapter 6, the only difference is that these AUCs should be in log scales (both x and y scales) in the context of contrast sensitivity.

Recall that `ACh_avg`

contains averaged contrast sensitivity for each spatial frequency across 51 observers for the first repetition.

```
## # A tibble: 24 × 5
## # Groups: Repetition [2]
## Repetition SpatialFreq avgSens stdErrSens stdDevSens
## <fct> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.25 12.6 0.743 5.31
## 2 1 0.35 14.7 0.777 5.55
## 3 1 0.49 18.6 0.993 7.09
## 4 1 0.68 23.7 1.22 8.71
## 5 1 0.94 28.7 1.36 9.68
## 6 1 1.31 32.0 1.40 9.99
## 7 1 1.83 32.4 1.43 10.2
## 8 1 2.54 29.6 1.39 9.93
## 9 1 3.54 24.3 1.22 8.70
## 10 1 4.93 18.1 0.960 6.85
## # ℹ 14 more rows
```

To compute the trapezoidal AUC, we can use the function `sm_trapz()`

.

`## [1] 2.081833`

The difference between `sm_trapz()`

and `sm_auc()`

is that `sm_trapz()`

automatically converts the `x`

and `y`

vectors into log10 scales. This default can be overcome when `logXY = FALSE`

. When, `logXY = FALSE`

, `sm_trapz()`

becomes identical to `sm_auc()`

. `sm_trapz()`

has been created as a separate function to create a convenient and straightforward workflow for those who wish to analyze the contrast sensitivity data.

In addition, to compute the trapezoidal AUC, we can use the function `sm_AULCSF()`

.

`## [1] 2.089264`

Notice that the AULCSF is very similar to AUC. This suggests that the CSF fit (as generated by `sm_CSF()`

) is very good. This can verified by computing \(R^2\) (coefficient of determination), which ranges from 0 (worst model fit) to 1 (best model fit), using `sm_r2()`

.

`## [1] 0.9672`

We see that \(R^2\) of the CSF fit is 0.9672, which is considered excellent. Note that `sm_r2()`

is a function specific for CSF fit only, and the `x`

and `y`

arguments should contain data in linear units, not log units.

To visualize what \(R^2\) captures, we first need to understand **residuals**, which is the difference or distance between the actual data and the predicted data from the given fit, such as the CSF.

```
ACh_avg1 %>% ggplot(aes(x = SpatialFreq, y = avgSens)) +
sm_CSF(color = sm_color("wine"), linetype = "dashed") +
geom_point(color = sm_color("darkred")) +
sm_hgrid() +
xlab("Spatial frequency (c/deg)") +
ylab("Averaged sensitivity") +
ggtitle("Repetition 1 (n=51)")
```

If the residual is very low, then the distance between the actual and predicted data should be close to 0. We see that the residual is low when the spatial frequency is between 3 and 10 c/deg. In this case, the general fit is very faithful to the original data, so the residual is low and \(R^2\) is high (close to 1). In sum, the lower the residuals, the higher the \(R^2\), which ranges from 0 to 1.

## 14.3 Calculating area and \(R^2\) of all subjects, groups and conditions

Now that we know how to calculate AUC, AULCSF and \(R^2\) for each subject, let’s calculate slopes of our entire dataset.

```
## # A tibble: 1,224 × 4
## Subject Repetition SpatialFreq Sensitivity
## <chr> <dbl> <dbl> <dbl>
## 1 S1 1 0.25 23.8
## 2 S2 1 0.25 18.0
## 3 S3 1 0.25 14.2
## 4 S4 1 0.25 5.14
## 5 S5 1 0.25 16.0
## 6 S6 1 0.25 10.9
## 7 S7 1 0.25 8.41
## 8 S8 1 0.25 10.5
## 9 S9 1 0.25 9.06
## 10 S10 1 0.25 3.57
## # ℹ 1,214 more rows
```

We see that there are 51 subjects total, each of which has completed two repetitions (i.e., conditions). So there are 102 slopes to calculate! Does that mean we need to use `sm_trapz()`

, `sm_AULCSF()`

and `sm_r2()`

102 times?

As the reader might expect from the previous chapters, the answer is **no**. **smCSF** has a functions `sm_trapz_list()`

, `sm_AULCSF_list()`

and `sm_r2_list()`

that return a data frame of AUCs, AULCSFs and \(R^2\)s, respectively. They works similarly to `sm_auc_list()`

and `sm_slope_list()`

from **smplot**.

`data`

= this argument requires the variable that stores the data frame. In our case, it is`ACh`

. It is recommended that the data are in linear units.`subjects`

= this argument requires the name of the column of the data frame that contains subjects. It must strings, ex.`'Subject'`

, not`Subject`

.`groups`

= this argument requires the name of the column of the data frame that contains each group. In this example, there is no group. An example would be`Group`

column that contains two groups:`Normal`

and`Amblyopia`

.`conditions`

= this argument requires name of the column of the data frame that contains each condition. In our example, the two conditions are`1`

and`2`

from the`Repetition`

column.`x`

= this argument requires the name of the column of the data frame that contains the x-axis points from which the AUC can be calculated. In our case, these are values from the`SpatialFreq`

column of`ACh`

. It must strings, ex.`'SpatialFreq'`

, not`SpatialFreq`

.`values`

= this argument requires the name of the column of the data frame that contains the actual data, which are the y-axis points from which the AUC can be calculated. In our case, it is the change in contrast balance ratio. It must strings, ex.`'Sensitivity'`

, not`Sensitivity`

.`logXY`

(for`sm_trapz_list()`

and`sm_AULCSF_list()`

) = thus argument’s default is`sm_trapz_list(logXY = TRUE)`

and`sm_AULCSF_list(logXY = TRUE)`

, so it computes the log10 units (`x`

and`values`

vector) of AUC and AULCSF. When`logXY = FALSE`

, it computes the linear units (`x`

and`values`

vector) AUC and AULCSF.

Let’s calculate the AUCs first. We can store the results from `sm_trapz_list()`

into a new variable. I will call the new variable `trapz_df`

.

```
trapz_df <- sm_trapz_list(
subjects = "Subject",
conditions = "Repetition",
x = "SpatialFreq",
values = "Sensitivity",
data = ACh
)
```

`## [1] "Trapezoid AUC = Sensitivity * SpatialFreq"`

```
## Subject Repetition AUC_Sensitivity
## 1 S1 1 2.299290
## 2 S1 2 2.032390
## 3 S2 1 2.246943
## 4 S2 2 2.215784
## 5 S3 1 1.973220
## 6 S3 2 2.204566
```

Likewise, we can store the results from `sm_AULCSF_list()`

in a new variable. I will call the new variable `aulcsf_df`

.

```
aulcsf_df <- sm_AULCSF_list(
subjects = "Subject",
conditions = "Repetition",
x = "SpatialFreq",
values = "Sensitivity",
data = ACh
)
```

`## [1] "AULCSF = Sensitivity * SpatialFreq"`

```
## Subject Repetition AULCSF
## 1 S1 1 2.307217
## 2 S1 2 2.038921
## 3 S2 1 2.254705
## 4 S2 2 2.222592
## 5 S3 1 1.979206
## 6 S3 2 2.214169
```

Finally, we can store the results from `sm_r2_list()`

in a new variable. I will call the new variable `r2`

_df`.

```
r2_df <- sm_r2_list(
subjects = "Subject",
conditions = "Repetition",
x = "SpatialFreq",
values = "Sensitivity",
data = ACh
)
```

`## [1] "R2 (0-worst, 1-best) for each subject"`

```
## Subject Repetition R2
## 1 S1 1 0.8605
## 2 S1 2 1.0000
## 3 S2 1 0.9086
## 4 S2 2 0.8731
## 5 S3 1 0.7505
## 6 S3 2 0.9947
```

We see that \(R^2\) is very good except for few observers in certain repetitions. Let’s see how many rows of the data frame contains \(R^2\) less than 0.75.

`## [1] 8`

```
## Subject Repetition R2
## 11 S6 1 0.5294
## 18 S9 2 0.6518
## 29 S15 1 0.5664
## 61 S31 1 0.6935
## 66 S33 2 0.6041
## 78 S39 2 0.5018
## 88 S44 2 0.7240
## 92 S46 2 0.5190
```

Out of 102 rows of the \(R^2\)s, only 8 of them contain poor \(R^2\). I think the contrast sensitivity function fit is acceptable to model the dataset of achromatic sensitivity in 51 normal observers.

## 14.4 Case Study

If one is interested in seeing whether a clinical treatment can improve the overall contrast sensitivity of the visually impaired, one can do so by comparing the area (AUC or AULCSF) before and after the treatment.

Let’s generate some fake data with `ACh`

. We will create a new column using `mutate()`

by creating a new column `BeforeAfter`

, which replaces `1`

with `Before`

and `2`

with `After`

to denote before and after receiving clinical treatment. Since `BeforeAfter`

is derived from the `Repetition`

column, it is a `<fct>`

column, as it is so with the `Repetition`

column. This new data frame is stored in the variable named `ACh1`

. We assume that the 51 subjects are visually impaired patients who received clinical treatment for visual improvement.

```
ACh$Repetition <- factor(ACh$Repetition)
ACh1 <- ACh %>%
mutate(BeforeAfter = fct_recode(Repetition,
"Before" = "1",
"After" = "2"
)) %>%
select(-Repetition)
```

Now, lets increase the values from the `Sensitivity`

column in rows that have `BeforeAfter == After`

using positive random numbers from a normal distribution that has a mean of `31`

and a standard deviation of `35`

using `rnorm()`

.

```
set.seed(33)
ind <- which(ACh1$BeforeAfter == "After")
ACh1[ind, ]$Sensitivity <- ACh1[ind, ]$Sensitivity + abs(rnorm(length(ind), 31, 35))
```

Now let’s calculate both the trapezoidal AUC and AULCSF and using `sm_trapz_list()`

and `sm_AULCSF_list()`

, both of which have defaults `logXY = TRUE`

.

```
trapz_df <- sm_trapz_list(
subjects = "Subject",
conditions = "BeforeAfter",
x = "SpatialFreq",
values = "Sensitivity",
data = ACh1
)
```

`## [1] "Trapezoid AUC = Sensitivity * SpatialFreq"`

```
## Subject BeforeAfter AUC_Sensitivity
## 1 S1 Before 2.299290
## 2 S1 After 2.867173
## 3 S2 Before 2.246943
## 4 S2 After 2.817482
## 5 S3 Before 1.973220
## 6 S3 After 2.856290
```

```
aulcsf_df <- sm_AULCSF_list(
subjects = "Subject",
conditions = "BeforeAfter",
x = "SpatialFreq",
values = "Sensitivity",
data = ACh1
)
```

`## [1] "AULCSF = Sensitivity * SpatialFreq"`

```
## Subject BeforeAfter AULCSF
## 1 S1 Before 2.307217
## 2 S1 After 2.874242
## 3 S2 Before 2.254705
## 4 S2 After 2.824033
## 5 S3 Before 1.979206
## 6 S3 After 2.849016
```

Now let’s plot a bar graph using `sm_bar()`

that compares both the trapezoidal AUCs and AULCSFs between before and after the visually impaired patients have received the clinical treatment.

```
trapz_df %>% ggplot(aes(
x = BeforeAfter, y = AUC_Sensitivity,
color = BeforeAfter
)) +
sm_bar(point.params = list(alpha = 0.3, size = 2)) +
scale_color_manual(values = sm_palette(2)) +
scale_y_continuous(limits = c(0, 4.5))
```

```
aulcsf_df %>% ggplot(aes(
x = BeforeAfter, y = AULCSF,
color = BeforeAfter
)) +
sm_bar(point.params = list(alpha = 0.3, size = 2)) +
scale_color_manual(values = sm_palette(2)) +
scale_y_continuous(limits = c(0, 4.5))
```

Both graphs suggest that the visual improvement has brought a large improvement of the overall contrast sensitivity in the patients who received the visual treatment. **Chapter 10 Basic Statistics** covers some methods to analyze data, and the reader can attempt to use the concepts in the previous chapter to analyze whether these differences from the simulated data are statistically significant.

Instead of focusing on the overall sensitivity of the observers, one can focus on observing whether the sensitivity of the high-spatial frequency range has improved after receiving the treatment. This, however, requires an understanding of the parameters that are used to fit the contrast sensitivity function. We will discuss these parameters in the next chapter.