eye

Tjebo Heeren

2021-03-08

See more with eye

library(eye)
library(eyedata)

Introduction

eye is dedicated to facilitate ophthalmic research, providing convenient application programming interfaces (API) for common tasks:

eye includes a visual acuity conversion chart.

Visual acuity

Pesky visual acuity notations are now a matter of the past. Convert between any of Snellen (meter/ feet/ decimal!), logMAR and ETDRS. The notation will be detected automatically and converted to the desired notation. For some more details see VA conversion. For entries with mixed notation, use va_mixed instead.

You can also decide to simply “clean” your VA vector with cleanVA(x). This will remove all entries that are certainly no VA.

va() (and of course, its wrappers) cleans and converts visual acuity notations (classes) between Snellen (decimal, meter and feet), ETDRS, and logMAR. Each class can be converted from one to another. va() will detect the class automatically based on specific rules detailed below. Calling va() without specifying the “to” argument will simply clean the visual acuity entries - any notations will be accepted, no plausibility checks yet performed. This is then bascially a wrapper around cleanVA.

It takes an (atomic) vector with visual acuity entries as the only required argument. The user can specify the original VA notation, but va will check that and ignore the argument if implausible.

Conversion steps

va() basically runs three main steps:

  1. Entry cleaning with clean_va()
  2. Notation detection with which_va()
  3. Plausibility checks with checkVA()
  4. Conversion with the S3 generic convertVA()

Cleaning

  1. NA are assigned to missing entries or strings representing such entries (“.”, "“,”{any number of spaces}“,”N/A“,”NA“,”NULL“,”-")
  2. Notation for qualitative entries is simplified (NPL becomes NLP, PL becomes LP).
  3. “plus” and “minus” from Snellen entries are converted:
    • if entry -2 to +2 : take same Snellen value
    • if < -2 : take Snellen value one line below
    • if >+2 : Snellen value one line above

Snellen are unfortunately often entered with “+/-”, which is a violation of psychophysical methods designed to assign one (!) unambiguous value to visual acuity, with non-arbitrary thresholds based on psychometric functions. Therefore, transforming “+/-” notation to actual results is in itself problematic and the below suggestion to convert it will remain an approximation to the most likely “true” result. Even more so, as the given conditions should work for charts with 4 or 5 optotypes in a line, and visual acuity is not always tested on such charts. Yet, I believe that the approach is still better than just omitting the letters or (worse) assigning a missing value to those entries.

If the argument smallstep = TRUE, the entries will be converted to logmar values (0.02 logmar for each optotype). This is based on the assumption of 5 optotypes in a row. This argument can be overriden with noplus = TRUE, ignoring the plus minus entries entirely and simply returning the nearest Snellen values.

Detection

Accepted VA formats / Plausibility checks [#plausibility]

  • Snellen fractions (meter/ feet) need to be entered as fraction with “/”. any fraction is allowed e.g. 3/60 and 2/200 will also be recognized.
  • ETDRS must be integer-equivalent between 0 and 100 (integer equivalent means, it can also be a character vector)
  • logMAR must be between -0.3 and 3.0
  • snellendec must be greater than 0 and smaller or equal to 2
  • Qualitative must be either of PL, LP, NLP, NPL, HM, CF (any case allowed)
  • Any element which is implausible / not recognized will be converted to NA

Conversion

Examples

## automatic detection of VA notation and converting to logMAR by default
x <- c(23, 56, 74, 58) ## ETDRS letters
to_logmar(x) # wrapper of va(x, to = "logmar")
#> From etdrs
#> [1] 1.24 0.58 0.22 0.54

## ... or convert to snellen
to_snellen(x) # wrapper of va(x, to = "snellen") 
#> From etdrs
#> [1] "20/320" "20/80"  "20/32"  "20/70"

## eye knows metric as well 
to_snellen(x, type = "m") 
#> From etdrs
#> [1] "6/96"  "6/24"  "6/9.6" "6/21"

## And the decimal snellen notation, so much loved in Germany
to_snellen(x, type = "dec") 
#> From etdrs
#> [1] "0.062" "0.25"  "0.625" "0.3"

## Remove weird entries and implausible entries depending on the VA choice
x <- c("NLP", "0.8", "34", "3/60", "2/200", "20/50", "  ", ".", "-", "NULL")

to_snellen(x)
#> From snellen. Could be snellen, logmar, snellendec, etdrs
#> 6x NA introduced for: 0.8, 34,   , ., -, NULL
#>  [1] "20/20000" NA         NA         "20/400"   "20/2000"  "20/50"   
#>  [7] NA         NA         NA         NA
to_snellen(x, from = "snellendec")
#> 8x NA introduced for: 34, 3/60, 2/200, 20/50,   , ., -, NULL
#>  [1] "20/20000" "20/25"    NA         NA         NA         NA        
#>  [7] NA         NA         NA         NA
to_snellen(x, from = "etdrs")
#> 8x NA introduced for: 0.8, 3/60, 2/200, 20/50,   , ., -, NULL
#>  [1] "20/20000" NA         "20/200"   NA         NA         NA        
#>  [7] NA         NA         NA         NA
to_snellen(x, from = "logmar")
#> 8x NA introduced for: 34, 3/60, 2/200, 20/50,   , ., -, NULL
#>  [1] "20/20000" "20/125"   NA         NA         NA         NA        
#>  [7] NA         NA         NA         NA

## "plus/minus" entries are converted to the most probable threshold (any spaces allowed) 
x <- c("20/200 - 1", "6/6-2", "20/50 + 3", "6/6-4", "20/33 + 4")
to_logmar(x)
#> From snellen
#> [1] 1.0 0.0 0.3 0.1 0.1

## or evaluating them as logmar values (each optotype equals 0.02 logmar)
to_logmar(x, smallstep = TRUE)
#> From snellen
#> [1] 1.02 0.04 0.34 0.08 0.14

## or you can also decide to completely ignore them (converting them to the nearest snellen value in the VA chart)
to_snellen(x, noplus = TRUE)
#> From snellen
#> [1] "20/200" "20/20"  "20/50"  "20/20"  "20/32"

recodeye

Makes recoding eye variables very easy.

The following codes are recognized:

If you have different codes, you can change the recognized strings with the eyecodes argument, which needs to be a list. But remember to put the strings for right eyes first, or pass a named list.

x <- c("r", "re", "od", "right", "l", "le", "os", "left")
recodeye(x)
#> [1] "r" "r" "r" "r" "l" "l" "l" "l"

## or with "both eyes"
x <- c(x, "both", "ou")
recodeye(x)
#>  [1] "r" "r" "r" "r" "l" "l" "l" "l" "b" "b"

## chose the resulting codes
recodeye(x, to = c("od", "os", "ou"))
#>  [1] "od" "od" "od" "od" "os" "os" "os" "os" "ou" "ou"

## Numeric codes 0:1/ 1:2 are recognized 
x <- 1:2
recodeye(x)
#> Eyes coded 1:2. Interpreting r = 1
#> [1] "r" "l"

## with weird missing values
x <- c(1:2, ".", NA, "", "    ")
recodeye(x)
#> Missing values and/or meaningless strings contained
#> Eyes coded 1:2. Interpreting r = 1
#> [1] "r" "l" NA  NA  NA  NA

## Or if you have weird codes for eyes
x <- c("alright", "righton", "lefty","leftover")

recodeye(x, eyecodes = list(r = c("alright","righton"), l = c("lefty","leftover")))
#> [1] "r" "r" "l" "l"

Counting patients and eyes

eyes offers a very simple tool for counting patients and eyes.

An important step in eyes is the guessing of the columns that identify patients and eyes. As for myop and of course blink, a specific column naming is required for a reliable automatic detection of patient and eye column(s) ( see Names and codes)

The arguments id and eye arguments overrule the name guessing for the respective columns.

Guessing

patient ID columns:

  • names can be in any case.
  • First, eyes is looking for names that contain both strings “pat” and “id” (the order doesn’t matter)
  • Next, it will look for columns that are plainly called “ID”
  • Last, it will search for all names that contain either “pat” or “id”

eye variable column:

  • names can be in any case.
  • eyes looks for columns that contain the string either “eye” or “eyes”
  • columns will full string “eye” or “eyes” will be given precedence

Counting

For counting eyes, eyes need to be coded in commonly used ways. You can use recodeye for very convenient recoding.

Report:

eyes also include a convenience function to turn the count into a text. This is intended for integration into rmarkdown reports, or for easy copy / pasting. eyes_to_string() parses the output of eyes into text under the hood. Arguments to eyes_to_string are passed via …:

eyestr is a shorthand for eyes(x, report = TRUE). The name was chosen because it’s a contraction of “eyes” and “strings” and it’s a tiny bit easier to type than “eyetxt”.

Use in rmarkdown

eyestr was designed with the use in rmarkdown in mind, most explicitly for the use inline:

We analysed `r eyestr(amd)`

will give:

We analysed 7802 patients.

The beauty of this report is of course that the numbers will always update with new data.

Examples

eyes(amd2)
#> patients     eyes    right     left 
#>     3357     3357     1681     1676

library(eyedata)
eyes(amd2)
#> patients     eyes    right     left 
#>     3357     3357     1681     1676

# Same as `eyes`, but as text for reports
eyestr(amd2)
#> [1] "3357 eyes of 3357 patients"

## Numbers smaller than or equal to 12 will be real English
eyestr(head(amd2, 100))
#> [1] "eleven eyes of eleven patients"

## Or, you can make all numbers english:
eyestr(amd2, english = "all")
#> [1] "three thousand three hundred and fifty-seven eyes of three thousand three hundred and fifty-seven patients"

## make first number capital letter
eyestr(head(amd2, 100), caps = TRUE)
#> [1] "Eleven eyes of eleven patients"

## you can have all numbers printed as numbers
eyestr(head(amd2, 100), english = "none")
#> [1] "11 eyes of 11 patients"

Reshape eye data

Out of convenience, data is often entered in a “wide” format: In eye research, there will be often two columns for the same variable, one column for each eye.

This may be a necessary data formal for specific questions.

However, “eye” is also variable (a dimension of your observation), and it can also be stored in a separate column. Indeed, in my experience R often needs eyes to be in a single column, with each other variable having their own dedicated column.

myop

Reshaping many such columns can be a daunting task, and myop() makes this easier. It will remove duplicate rows, and pivot the eye variable to one column and generate a single column for each variable, thus shaping the data for specific types of analysis. For example, eight columns that store data of four variables for right and left eyes will be pivoted to 5 columns (one eye column and four further variable columns)). See also Examples.

As with eyes(), myop() requires a specific data format. See names and codes If there is already a column called “eye” or “eyes”, myop will not make any changes - because the data is then already assumed to be in long format.

If there still are variables spread over two columns for right and left eyes, then this is an example of messy data. A solution would be to remove or simply rename the “eye” column and then let myop do the work. However, you need to be very careful in those cases if resulting data frame is plausible.

Learn about tidy data.

Make myop work

myop will work reliably if you adhere to the following:

  1. Common codes for eyes:
    • Right eyes: “r”, “re”, “od”, “right”
    • Left eyes: “l”, “le”, “os”, “left”
  2. strings for eyes need to be separated by period or underscores. (Periods will be replaced by underscores).
  3. Any order of substrings is allowed:
    • Will work: “va_r”, “right_morningpressure”, “night_iop.le”, “gat_os_postop”
    • Will fail: “VAr”, “rightmorningPressure”, “night_IOPle”, “gatOSpostop”

An exception is when there is only one column for each eye. Then the column names can consist of “eye strings” only. In this case, the argument var will be used to name the resulting variable.

If there are only eye columns in your data (should actually not happen), myop will create identifiers by row position.

Please always check the result for plausibility. Depending a lot on how the data was entered, the results could become quite surprising. There is basically a nearly infinite amount of possible combinations of how to enter data, and it is likely that myop will not be able to deal with all of them.

myop under the hood

myop() basically runs three main steps:

  1. Removing duplicates
  2. Rename data names with myop_rename() and sort_substr():
    • Replacing “.” by "_"
    • Re-arranging and recoding substrings in a way that strings for eyes always appear at first position. They will be recoded to “r” and “l”
  3. Myopization: The actual work is done with myopizer() and myop_pivot() and itself consists of three steps.
    • All columns with an eye string at first position will be selected pivoted to two long colums (key and value) using tidyr::pivot_longer.
    • The key column will be split by position into an eye column and a variable column.
    • The variable and value columns will be pivoted wide again with tidyr::pivot_wider.
## the variable has not been exactly named, (but it is probably IOP data), 
## you can specify the dimension with the var argument
wide1 <- data.frame(id = letters[1:3],  r = 11:13 , l = 14:16)
myop(wide1, var = "iop")
#> # A tibble: 6 x 3
#>   id    eye   iop  
#>   <chr> <chr> <chr>
#> 1 a     r     11   
#> 2 a     l     14   
#> 3 b     r     12   
#> 4 b     l     15   
#> 5 c     r     13   
#> 6 c     l     16

## If the dimension is already part of the column names, this is not necessary. 
iop_wide <- data.frame(id = letters[1:3], iop_r = 11:13, iop_l = 14:16)
myop(iop_wide)
#> # A tibble: 6 x 3
#>   id    eye   iop  
#>   <chr> <chr> <chr>
#> 1 a     r     11   
#> 2 a     l     14   
#> 3 b     r     12   
#> 4 b     l     15   
#> 5 c     r     13   
#> 6 c     l     16

## Mildly messy data frame with several variables spread over two columns:
wide_df <- data.frame(
  id = letters[1:4], 
  surgery_right = c("TE", "TE", "SLT", "SLT"),
  surgery_left = c("TE", "TE", "TE", "SLT"),
  iop_r_preop = 21:24, iop_r_postop = 11:14,
  iop_l_preop = 31:34, iop_l_postop = 11:14, 
  va_r_preop = 41:44, va_r_postop = 45:48,
  va_l_preop = 41:44, va_l_postop = 45:48
)

## myop deals with this in a breeze:
myop(wide_df)
#> # A tibble: 8 x 7
#>   id    eye   surgery iop_preop iop_postop va_preop va_postop
#>   <chr> <chr> <chr>   <chr>     <chr>      <chr>    <chr>    
#> 1 a     r     TE      21        11         41       45       
#> 2 a     l     TE      31        11         41       45       
#> 3 b     r     TE      22        12         42       46       
#> 4 b     l     TE      32        12         42       46       
#> 5 c     r     SLT     23        13         43       47       
#> 6 c     l     TE      33        13         43       47       
#> 7 d     r     SLT     24        14         44       48       
#> 8 d     l     SLT     34        14         44       48

hyperop

Basically the opposite of myop() - a slightly intelligent wrapper around tidyr::pivot_longer() and tidyr::pivot_wider(). Will find the eye column, unify the codes for the eyes (all to “r” and “l”) and pivot the columns wide, that have been specified in “cols”. Again, good names and tidy data always help!

The cols argument takes a tidyselection. Read about tidyselection

myop_df <- myop(wide_df)
hyperop(myop_df, cols = matches("va|iop"))
#> # A tibble: 5 x 10
#>   id    surgery r_iop_preop r_iop_postop r_va_preop r_va_postop l_iop_preop
#>   <chr> <chr>   <chr>       <chr>        <chr>      <chr>       <chr>      
#> 1 a     TE      21          11           41         45          31         
#> 2 b     TE      22          12           42         46          32         
#> 3 c     SLT     23          13           43         47          <NA>       
#> 4 c     TE      <NA>        <NA>         <NA>       <NA>        33         
#> 5 d     SLT     24          14           44         48          34         
#> # … with 3 more variables: l_iop_postop <chr>, l_va_preop <chr>,
#> #   l_va_postop <chr>

Names and codes

eye works smoother with tidy data, and with good names (any package does, really!)

Tidy data

The basic principle of tidy data is: one column for each dimension and one row for each observation.

Learn more about tidy data.

This chapter explains how you can improve names and codes so that eye will work like a charm.

How do I rename columns in R?

When I started with R, I found it challenging to rename columns and I found the following methods very helpful:

I’ve got a data frame with unfortunate names:

name_mess <- data.frame(name = "a", oculus = "r", eyepressure = 14, vision = 0.2)
names(name_mess)
#> [1] "name"        "oculus"      "eyepressure" "vision"

I can rename all names easily:

names(name_mess) <- c("patID", "eye", "IOP", "VA")
names(name_mess)
#> [1] "patID" "eye"   "IOP"   "VA"

To rename only specific columns, even if you are not sure about their exact position:

## if you only want to rename one or a few columns: 
names(name_mess)[names(name_mess) %in% c("name", "vision")] <- c("patID", "VA")
names(name_mess)
#> [1] "patID"       "oculus"      "eyepressure" "VA"

For even more methods, I found those two threads on Stackoverflow very helpful:

Tips and rules for naming:

  1. Don’t be too creative with your names!
  2. Use common coding:
    • eyes: “r”, “re”, “od”, “right” - or numeric coding r:l = 0:1 or 1:2
    • Visual acuity: “VA”, “BCVA”, “Acuity”
    • Intraocular pressure: “IOP”, “GAT”, “NCT”, “pressure”
    • Patient identifier: “pat”, “patient”, “ID” (ideally both: “patientID” or “patID”)
  3. Column names:
    • No spaces!
    • Do not use numeric coding for eyes in column names
    • Separate eye and VA and IOP codes with underscores (“bcva_l_preop”, “VA_r”, “left_va”, “IOP_re”)
    • Keep names short
    • Don’t use underscores when you don’t need to: Consider each section divided by an underscore as a relevant characteristic of your variable. E.g., “preop” instead of “pre_op”, or simply “VA” instead of “VA_ETDRS_Letters”

Name examples

Good names (eye will work nicely)

## right and left eyes have common codes
## information on the tested dimension is included ("iop")
## VA and eye strings are separated by underscores
## No unnecessary underscores.
names(wide_df)
#>  [1] "id"            "surgery_right" "surgery_left"  "iop_r_preop"  
#>  [5] "iop_r_postop"  "iop_l_preop"   "iop_l_postop"  "va_r_preop"   
#>  [9] "va_r_postop"   "va_l_preop"    "va_l_postop"

names(iop_wide) 
#> [1] "id"    "iop_r" "iop_l"

OK names (eye will work)

## Id and Eye are common names, there are no spaces
## VA is separated from the rest with an underscore
## BUT: 
## The names are quite long 
## There is an unnecessary underscore (etdrs are always letters). Better just "VA"
c("Id", "Eye", "FollowupDays", "BaselineAge", "Gender", "VA_ETDRS_Letters", 
"InjectionNumber")
#> [1] "Id"               "Eye"              "FollowupDays"     "BaselineAge"     
#> [5] "Gender"           "VA_ETDRS_Letters" "InjectionNumber"

## All names are commonly used (good!)
## But which dimension of "r"/"l" are we exactly looking at? 
c("id", "r",  "l")
#> [1] "id" "r"  "l"

Bad names (eye will fail)

## VA/IOP not separated with underscore
## `eye` won't be able to recognize IOP and VA columns
c("id", "iopr", "iopl", "VAr", "VAl")
#> [1] "id"   "iopr" "iopl" "VAr"  "VAl"

## A human may think this is clear
## But `eye` will fail to understand those variable names
c("person", "goldmann", "vision")
#> [1] "person"   "goldmann" "vision"

## Not even clear to humans
c("var1", "var2", "var3")
#> [1] "var1" "var2" "var3"

Reveal common statistics

reveal() offers a simple API to show common summary statistics for all numeric columns of your data frame. reveal() is basically a slightly complicated wrapper around mean(), sd(), length(), min() and max() (with na.rm = TRUE and length() counting only non-NA values).

It is not really intended to replace other awesome data exploration packages / functions such as skimr::skim, and it will likely remain focussed on summarizing numerical data only.

It uses an S3 generic under the hood with methods for atomic vectors, data frames, and lists of either atomic vectors or data frames. Character vectors will be omitted (and it should give a warning that it has done so).

reveal() takes the grouping argument by and it returns vector for atomic vectors or a data frame for lists.

Examples

clean_df <- myop(wide_df)
reveal(clean_df)
#>          var mean  sd n min max
#> 1  iop_preop 27.5 5.5 8  21  34
#> 2 iop_postop 12.5 1.2 8  11  14
#> 3   va_preop 42.5 1.2 8  41  44
#> 4  va_postop 46.5 1.2 8  45  48

reveal(clean_df, by = "eye")
#>   eye        var mean  sd n min max
#> 1   l  iop_preop 32.5 1.3 4  31  34
#> 2   l iop_postop 12.5 1.3 4  11  14
#> 3   l   va_preop 42.5 1.3 4  41  44
#> 4   l  va_postop 46.5 1.3 4  45  48
#> 5   r  iop_preop 22.5 1.3 4  21  24
#> 6   r iop_postop 12.5 1.3 4  11  14
#> 7   r   va_preop 42.5 1.3 4  41  44
#> 8   r  va_postop 46.5 1.3 4  45  48

reveal(clean_df, by = c("eye", "surgery"))
#>    eye surgery        var mean  sd n min max
#> 1    l     SLT  iop_preop 34.0  NA 1  34  34
#> 2    l     SLT iop_postop 14.0  NA 1  14  14
#> 3    l     SLT   va_preop 44.0  NA 1  44  44
#> 4    l     SLT  va_postop 48.0  NA 1  48  48
#> 5    r     SLT  iop_preop 23.5 0.7 2  23  24
#> 6    r     SLT iop_postop 13.5 0.7 2  13  14
#> 7    r     SLT   va_preop 43.5 0.7 2  43  44
#> 8    r     SLT  va_postop 47.5 0.7 2  47  48
#> 9    l      TE  iop_preop 32.0 1.0 3  31  33
#> 10   l      TE iop_postop 12.0 1.0 3  11  13
#> 11   l      TE   va_preop 42.0 1.0 3  41  43
#> 12   l      TE  va_postop 46.0 1.0 3  45  47
#> 13   r      TE  iop_preop 21.5 0.7 2  21  22
#> 14   r      TE iop_postop 11.5 0.7 2  11  12
#> 15   r      TE   va_preop 41.5 0.7 2  41  42
#> 16   r      TE  va_postop 45.5 0.7 2  45  46

Calculate age

This is a simple function and should not require much explanation. However, it may be noteworthy to mention the subtle distinction of periods and durations, which are an idiosyncrasy of time measurements and well explained in this thread.

Examples

dob <- c("1984-10-16", "2000-01-01")

## If no second date given, the age today
getage(dob)
#> [1] 36.4 21.2

## If the second argument is specified, the age until then
getage(dob, "2000-01-01")                                                    
#> [1] 15.2  0.0

Important notes

I do not assume responsability for your data or analysis. Please always keep a critical mind when working with data - if you do get results that seem implausible, there may be a chance that the data is in an unfortunate shape for which eye may not be suitable.

VA conversion chart

This chart is included in the package as va_chart

Snellen feet Snellen meter Snellen decimal logMAR ETDRS Categories
20/20000 6/6000 0.001 3 0 NLP
20/10000 6/3000 0.002 2.7 0 LP
20/4000 6/1200 0.005 2.3 0 HM
20/2000 6/600 0.01 1.9 2 CF
20/800 6/240 0.025 1.6 5 NA
20/630 6/190 0.032 1.5 10 NA
20/500 6/150 0.04 1.4 15 NA
20/400 6/120 0.05 1.3 20 NA
20/320 6/96 0.062 1.2 25 NA
20/300 6/90 0.067 1.18 26 NA
20/250 6/75 0.08 1.1 30 NA
20/200 6/60 0.1 1.0 35 NA
20/160 6/48 0.125 0.9 40 NA
20/125 6/38 0.16 0.8 45 NA
20/120 6/36 0.167 0.78 46 NA
20/100 6/30 0.2 0.7 50 NA
20/80 6/24 0.25 0.6 55 NA
20/70 6/21 0.29 0.54 58 NA
20/63 6/19 0.32 0.5 60 NA
20/60 6/18 0.33 0.48 61 NA
20/50 6/15 0.4 0.4 65 NA
20/40 6/12 0.5 0.3 70 NA
20/32 6/9.6 0.625 0.2 75 NA
20/30 6/9 0.66 0.18 76 NA
20/25 6/7.5 0.8 0.1 80 NA
20/20 6/6 1.0 0.0 85 NA
20/16 6/5 1.25 -0.1 90 NA
20/15 6/4.5 1.33 -0.12 91 NA
20/13 6/4 1.5 -0.2 95 NA
20/10 6/3 2.0 -0.3 100 NA

Acknowledgements

Resources

References

Beck, Roy W, Pamela S Moke, Andrew H Turpin, Frederick L Ferris, John Paul SanGiovanni, Chris A Johnson, Eileen E Birch, et al. 2003. “A Computerized Method of Visual Acuity Testing.” American Journal of Ophthalmology 135 (2). Elsevier BV: 194–205. https://doi.org/10.1016/s0002-9394(02)01825-1.

Gregori, Ninel Z, William Feuer, and Philip J Rosenfeld. 2010. “Novel Method for Analyzing Snellen Visual Acuity Measurements.” Retina 30 (7). Ovid Technologies (Wolters Kluwer Health): 1046–50. https://doi.org/10.1097/iae.0b013e3181d87e04.

Holladay, Jack T. 2004. “Visual Acuity Measurements.” Journal of Cataract and Refractive Surgery 30 (2): 287–90. https://doi.org/10.1016/j.jcrs.2004.01.014.

Schulze-Bonsel, Kilian, Nicolas Feltgen, Hermann Burau, Lutz Hansen, and Michael Bach. 2006. “Visual Acuities ‘Hand Motion’ and ‘Counting Fingers’ Can Be Quantified with the Freiburg Visual Acuity Test.” Investigative Ophthalmology & Visual Science 47 (3): 1236–40. https://doi.org/10.1167/iovs.05-0981.