`library(singcar)`

The aim of the R package `singcar`

is to provide and encourage usage of
appropriate statistical methods for comparing a case against a control sample.
For instance, they may commonly be done in a neuropsychological context, in
which an individual has incurred a specific brain injury and we wish to test
whether this damage has led to an impairment of some cognitive function and
whether two different functions are dissociable. For many functions there is
normed data available which the patient can be compared against directly.
However, when this is not possible a control sample estimating the population,
against which we wish to compare the patient, must be used. Both frequentist and
Bayesian methods have been developed to do this providing transparent control
over Type I errors, first and foremost by John Crawford and Paul Garthwaite
(Crawford et al., 2011; Crawford & Garthwaite, 2002, 2007, 2005; Crawford & Howell, 1998). It is these methods that
`singcar`

implements. Due to the somewhat overlooked issue of Type II errors
power calculators for these tests are also provided. Although the canonical
applications for these tests are in Cognitive Neuropsychology or Clinical
Neuropsychology, they are potentially applicable to any circumstance in which a
measure taken from a single individual is to be compared against data from a
normative sample (i.e. a control group). It should be noted that these
statistical methods could also be applied as a general method of outlier
detection in small samples.

You can install the developmental version of `singcar`

by running the following:

```
install.packages("devtools")
library("devtools")
install_github("jorittmo/singcar")
library("singcar")
```

The package comes with the dataset `size_weight_illusion`

, a neuropsychological
dataset from an investigation of the size-weight illusion in DF, a patient with
visual form agnosia following following bilateral lesions to the lateral
occipital complex (Hassan et al., 2020). It was investigated whether
DF experienced visual size-weight illusion to the same extent as controls (n = 28)
and whether visual and kinesthetic size-weight illusion could be dissociable.
Below follows examples of how to analyse this dataset using the tests provided
in `singcar`

.

If we want to assess whether DF has an impairment compared to controls on visual size-weight illusion we can test this using a modified two-sample t-test, called TD (test of deficit: Crawford & Howell, 1998).

```
# Extracting scores from the visual size-weight illusion from size_weight_illusion
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls
CON_V_SWI
TD(case = DF_V_SWI, controls = CON_V_SWI, conf_int = TRUE)
#>
#> Test of Deficit
#>
#> data: Case = 0.03, Controls (m = 0.16, sd = 0.08, n = 28)
#> t = -1.7243, df = 27, p-value = 0.04804
#> alternative hypothesis: true difference between case and controls is less than 0
#> sample estimates:
#> Std. case score (Z-CC), 95% CI [-2.34, -1.15]
#> -1.754857
#> Proportion below case (%), 95% CI [0.95, 12.47]
#> 4.804003
```

This can similarly be tested with a Bayesian version of the same test, yielding approximately (since this test is based on MCMC methods) the same output (Crawford & Garthwaite, 2007).

```
# Extracting scores from the visual size-weight illusion from size_weight_illusion
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls
CON_V_SWI
BTD(case = DF_V_SWI, controls = CON_V_SWI)
#>
#> Bayesian Test of Deficit
#>
#> data: Case = 0.03, Controls (m = 0.16, sd = 0.08, n = 28)
#> df = 27, p-value = 0.04833
#> alternative hypothesis: true difference between case and controls is less than 0
#> sample estimates:
#> Std. case score (Z-CC), 95% CI [-2.34, -1.15]
#> -1.754857
#> Proportion below case (%), 95% CI [0.96, 12.57]
#> 4.832849
```

If the control sample for a study is not appropriately matched to the case on variables such as e.g. age or education level it is appropriate to use tests that account for this by allowing for the inclusion of covariates. Including theoretically sound covariates is often a good idea. To do this Crawford et al. (2011) extended their Bayesian verison of the TD. This test assess the patient on the task of interest by essentially comparing him/her to the controls with the same score on the covariate.

```
# Extracting scores from the visual size-weight illusion from size_weight_illusion
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls
CON_V_SWI
# Extracting the coviariate below
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "YRS"] # Patient
DF_age <- size_weight_illusion[size_weight_illusion$PPT != "DF", "YRS"] # Controls
CON_age
BTD_cov(case_task = DF_V_SWI, case_covar = DF_age, control_task = CON_V_SWI,
control_covar = CON_age, iter = 100)
#>
#> Bayesian Test of Deficit with Covariates
#>
#> data: Case = 0.03, Controls (m = 0.16, sd = 0.08, n = 28)
#> df = 26, p-value = 0.05838
#> alternative hypothesis: true difference between case and controls is less than 0
#> sample estimates:
#> Std. case score (Z-CCC), 95% CI [-2.29, -1.08]
#> -1.749556
#> Proportion below case (%), 95% CI [1.09, 14.03]
#> 5.837549
```

If we want to assess whether DF has a dissociation between two functions we can
use a modified paired samples t-test to assess the size of the difference
between the case scores from the two tasks to the distribution of differences
between the tasks in the controls. This can however only be done directly using
the t-distribution if the tasks are measured on the same scale and is called the
unstandardised difference test (UDT: Crawford & Garthwaite, 2005). In the
`size_weight_illusion`

dataset it is possible to use this test to whether
patient DF exhibits a dissociation between visual size-weight illusion and
kinesthetic size-weight illusion because the visual and kinaesthetic conditions
are parallel versions of the same task, with different sensory cues. This would
be done as shown below:

```
# Extracting scores from the visual size-weight illusion from size_weight_illusion
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls
CON_V_SWI
# Extracting scores from the kinesthetic size-weight illusion from size_weight_illusion
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient
DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls
CON_K_SWI
UDT(case_a = DF_V_SWI, case_b = DF_K_SWI, controls_a = CON_V_SWI, controls_b = CON_K_SWI)
#>
#> Unstandardised Difference Test
#>
#> data: Case A: 0.03, B: 0.10, Ctrl. A (m, sd): (0.16, 0.08), B: (0.18, 0.10)
#> t = -0.6667, df = 27, p-value = 0.5106
#> alternative hypothesis: true difference between tasks is not equal to 0
#> sample estimates:
#> Std. case score, task A (Z-CC)
#> -0.13647439
#> Std. case score, task B (Z-CC)
#> -0.07931545
#> Std. task discrepancy (Z-DCC), 95% CI [-1.53, -0.59]
#> -1.06478887
#> Proportion below case (%), 95% CI [6.35, 27.68]
#> 25.53097678
```

Most often this is not possible because we wish to estimate abnormality of discrepancy on tasks that are not comparable. So otherwise, that is if the scores must be standardised to be comparable, a statistic that approximates the t-distribution has been developed and should be used (the revised standardised difference test RSDT: Crawford & Garthwaite, 2005). The visual and kinesthetic size-weight illusion will be used for illustrative purposes here as well:

```
# Extracting scores from the visual size-weight illusion from size_weight_illusion
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls
CON_V_SWI
# Extracting scores from the kinesthetic size-weight illusion from size_weight_illusion
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient
DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls
CON_K_SWI
RSDT(case_a = DF_V_SWI, case_b = DF_K_SWI, controls_a = CON_V_SWI, controls_b = CON_K_SWI)
#>
#> Revised Standardised Difference Test
#>
#> data: Case A: 0.03, B: 0.10, Ctrl. A (m, sd): (0.16, 0.08), B: (0.18, 0.10)
#> approx. abs. t = 1.015, df = 27, p-value = 0.3191
#> alternative hypothesis: true difference between tasks is not equal to 0
#> sample estimates:
#> Std. case score, task A (Z-CC) Std. case score, task B (Z-CC)
#> -1.7548574 -0.7836956
#> Std. task discrepancy (Z-DCC) Proportion below case (%)
#> -1.0647889 15.9560625
```

A Bayesian version of this test was also developed (Crawford & Garthwaite, 2005),
however, unlike `TD`

and `BTD`

the `RSDT`

and `BSDT`

(Bayesian standardised
difference test) differ somewhat and `BSDT`

has been shown to keep a better
control of Type I errors if a patient exhibits extreme deficits on both tasks of
interest. Therefore the `BSDT`

is recommended above `RSDT`

. The usage of the two
R functions is very similar. Since the `BSDT`

is based on MCMC methods it can be
quite computationally intensive, depending on the number of iterations you choose.

```
# Extracting scores from the visual size-weight illusion from size_weight_illusion
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls
CON_V_SWI
# Extracting scores from the kinesthetic size-weight illusion from size_weight_illusion
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient
DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls
CON_K_SWI
BSDT(case_a = DF_V_SWI, case_b = DF_K_SWI, controls_a = CON_V_SWI, controls_b = CON_K_SWI, iter = 1000)
#>
#> Bayesian Standardised Difference Test
#>
#> data: Case A: 0.03, B: 0.10, Ctrl. A (m, sd): (0.16,0.08), B: (0.18,0.10)
#> df = 26, p-value = 0.3158
#> alternative hypothesis: true difference between tasks is not equal to 0
#> sample estimates:
#> Std. case score, task A (Z-CC)
#> -1.7548574
#> Std. case score, task B (Z-CC)
#> -0.7836956
#> Std. discrepancy (Z-DCC), 95% CI [-1.69, -0.47]
#> -1.0647889
#> Proportion below case (%), 95% CI [4.52, 31.95]
#> 15.7884713
```

Just as for `BTD`

a version of `BSDT`

allowing for
covariates has been developed. This test assess the patient on the discrepancy
between the tasks of interest by essentially comparing him/her to the controls
with the same score on the covariate.

```
# Extracting scores from the visual size-weight illusion from size_weight_illusion
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls
CON_V_SWI
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient
DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls
CON_K_SWI
# Extracting the coviariate below
<- size_weight_illusion[size_weight_illusion$PPT == "DF", "YRS"] # Patient
DF_age <- size_weight_illusion[size_weight_illusion$PPT != "DF", "YRS"] # Controls
CON_age
BSDT_cov(case_tasks = c(DF_V_SWI, DF_K_SWI ), case_covar = DF_age,
control_tasks = cbind(CON_V_SWI, CON_K_SWI), control_covar = CON_age, iter = 1000)
#>
#> Bayesian Standardised Difference Test with Covariates
#>
#> data: Case A = 0.03, B = 0.10, Ctrl. (m, sd) A: (0.16,0.08), B: (0.18,0.10)
#> df = 25, p-value = 0.3282
#> alternative hypothesis: true difference between tasks is not equal to 0
#> sample estimates:
#> Std. case score, task A (Z-CC)
#> -1.754857
#> Std. case score, task B (Z-CC)
#> -0.783696
#> Std. discrepancy (Z-DCCC), 95% CI [-1.70, -0.42]
#> -1.064152
#> Proportion below case (%), 95% CI [4.45, 33.80]
#> 16.410000
```

All of the functions above can also take summary (mean, sd, control sample size) data as input.

A further capacity of `singcar`

is that it can be used to calculate power for
for these single case-control comparisons. Calculations for all Bayesian tests
and `RSDT`

are simulation based and (especially the tests with covariates) can
be computationally intense. Calculators for `TD`

and `UDT`

(unstandardised
difference test) are exact (their power functions have been derived
analytically) and can both be used to find a specific sample size given a
desired power. For the other calculators all parameters must be given. Means and
standard deviations for the control population are at default set to 0 and 1
meaning that the case value will be interpreted as differences from the mean in
standard deviations, these parameter values can be changed as you like. Examples
are given below:

```
TD_power(case = -2, power = 0.8, mean = 0, sd = 1, alternative = "two.sided")
#> Power (0.443) will not increase more than 0.5% per participant for n > 16
#> n power
#> 1 16 0.4428042
TD_power(case = 70, sample_size = 10, mean = 100, sd = 15, alternative = "less", alpha = 0.1)
#> [1] 0.7039033
RSDT_power(case_a = 70, case_b = 20, mean_a = 100, mean_b = 25, sd_a = 15, sd_b = 10, sample_size = 10)
#> [1] 0.2361
# Takes long time to compute therefore iterations and number of simulations are low. Iter corresponds
# to number of simulations in BTD_cov, nsim to the number of simulations in the power calculator.
BTD_cov_power(case = -2, case_cov = 0, control_task = c(0, 1),
control_covar = c(0, 1), cor_mat = diag(2), sample_size = 10, nsim = 50, iter = 50)
#> [1] 0.56
```

The main functions of `singcar`

are hitherto:

Frequentist tests:

`TD()`

: The test of deficits. Used to test for abnormality on a single variate.`UDT()`

: The unstandardised difference test. Used to test for discrepancy between two variates measured on the same scale.`RSDT()`

: The revised standardised difference test. Used to test for discrepancy between two variates measured on different (or the same) scale.

Bayesian tests:

`BTD()`

: Bayesian test of deficit. Used to test for abnormality on a single variate.`BSDT()`

: Bayesian standardised difference test. Used to test for discrepancy between two variates measured on the same scale or different scales (depending on the`unstandardised`

argument).`BTD_cov()`

: Bayesian test of deficit with covariates. Used to test for abnormality on a single variate conditioned on some covariate such as e.g. age or education level.`BSDT_cov()`

: Bayesian standardised difference test with covariates. Used to test for discrepancy between two variates conditioned on some covariates such as e.g. age or education level.

Power calculators:

`TD_power()`

: Calculates exact power given sample size or necessary sample size for desired power using analytical methods for the test of deficit.`BTD_power()`

: Calculates approximate power given sample size using Monte Carlo simulations for the Bayesian test of deficit.`BTD_cov_power()`

: Calculates approximate power given sample size using Monte Carlo simulations for the Bayesian test of deficit with covariates.`UDT_power()`

: Calculates exact power given sample size or necessary sample size for desired power using analytical methods for the unstandardised difference test.`RSDT_power()`

: Calculates approximate power given sample size using Monte Carlo simulations for the revised standardised difference test.`BSDT_power()`

: Calculates approximate power given sample size using Monte Carlo simulations for the Bayesian standardised difference test.`BSDT_cov_power()`

: Calculates approximate power given sample size using Monte Carlo simulations for the Bayesian standardised difference test with covariates.

The aim of this section is not to provide an exhaustive mathematical understanding of the formulas used but rather a conceptual understanding. This is written from a neuropsychological perspective. However, that does not make the tests exclusive to such an application.

In the latter part of the 90’s Crawford and colleagues started developing statistical tests to use for evaluating single cases that were compared to normative control samples. Typically, the prior methods used treated the distribution estimated by the the control group as if the sample statistics were the population parameters. That is, the estimated distribution was treated as a standard normal distribution from which abnormality of the case score was estimated by:

\[\begin{equation} z = \frac{x^* - \overline{x}}{\sqrt{s^2}} \tag{1} \end{equation}\]

This is similar to the familiar z-formula but here \(x^*\), \(\overline{x}\) and \(s^2\) is the case score, sample mean and sample variance respectively. \(\overline{x}\) and \(s^2\) is plugged in directly as the population parameters \(\mu\) and \(\sigma^2\) in the normal z-formula. The p-value obtained from the z-value would then be treated as the estimation of the case’s abnormality. This is problematic because the sampling distribution of \(s^2\) is right skewed for small sample sizes. This means that underestimation of \(s^2\) is more probable than overestimation and hence the z-value would often be larger than it should, resulting in an overestimation of the abnormality and inflation of Type I errors (claiming that there is an effect when, in fact, there is not) (Crawford & Howell, 1998).

With a similar logic as in (1), Payne & Gwynne Jones (1957) developed a method for assessing abnormally large discrepancies between two tasks. I.e. a test that estimates the proportion of the control population that would exhibit a greater discrepancy than the case, as seen in (2).

\[\begin{equation} z_{disc} = \frac{(z^*_a - z^*_b) - {(\overline{z}_a - \overline{z}_b)} }{\sqrt{2-2r_{ab}}} = \frac{(z^*_a - z^*_b)}{\sqrt{2-2r_{ab}}} \tag{2} \end{equation}\]

Where \(z^*_a\) and \(z^*_b\) are the standardised case scores on task A and B respectively, \(\overline{z}_a\) and \(\overline{z}_b\) the means from the sample on the two tasks (which both equates 0 because of standardisation) and \(r_{ab}\) the correlation between the two tasks calculated from the sample scores. However, this test suffers from the same problem mentioned above and would overestimate the abnormality of the task discrepancies.

A different approach to comparing a single observation to the mean of a sample was proposed by Sokal & Rohlf (1981) (p. 227) and popularised within neuropsychology by Crawford & Howell (1998). Here the t-distribution (with its fatter tails) is utilised to account for the underestimation of the sample variance. The approach is a modified two samples t-test where the case simply is treated as a sample of size 1. The degrees of freedom for this distribution is \(n + 1 - 2 = n - 1\).

\[\begin{equation} t_{n-1} = \frac{X^* - \overline{X}}{s \sqrt{\frac{n + 1}{n}}} \tag{3} \end{equation}\]

This test of deficit (TD) has been shown to not exceed the specified error rate
\(\alpha\) unlike other similar tests (Crawford et al., 2004, 2009; Crawford & Garthwaite, 2012). Together with its
simplicity this makes it a superior choice over many other ways of
detecting outliers in small samples. One of its main advantages is that it
provides the researcher with an *unbiased* point estimate of the abnormality of
the case.

Crawford et al. (1998) extended this to Payne & Gwynne Jones (1957) test of task
discrepancy or with Crawford et al. (1998)'s denotation: ‘difference’^{1}. I.e. they devised a test that treated sample
estimations as statistics rather than population parameters for dissociations as well, seen
in (4).
\[\begin{equation}
t_{n-1} = \frac{(z^*_a - z^*_b) {- (\overline{z}_a - \overline{z}_b)} }{\sqrt{(2-2r_{ab})(\frac{n+1}{n})}} = \frac{(z^*_a - z^*_b)}{\sqrt{(2-2r_{ab})(\frac{n+1}{n})}}
\tag{4}
\end{equation}\]
Unfortunately the standardised task scores of the case \(z^*_a\) and \(z^*_b\)
suffer from the same problem described for (1) and Type I errors
would again be inflated. However, standardisation
of the scores is only necessary if the two tasks are measured on different scales.
If they are measured on the same the test holds and we have the unstandardised
difference test (UDT):

\[\begin{equation} t_{UDT_{n-1}} = \frac{(x^*_a - \overline{x}_a) - (x^*_b - \overline{x}_b) }{\sqrt{(s^2_a +s^2_b -2s_a s_br_{ab})(\frac{n+1}{n})}} \tag{5} \end{equation}\]

The denominator in the (5) collapse to the denominator in (4) since \(s_a^2\) and \(s_b^2\) become 1 after standardisation. However, since assessment of task discrepancy between tasks measured on different scales is common, a test that could take standardised scores but still account for the skewness in the sampling distribution of the sample variance was needed. In Garthwaite & Crawford (2004) the authors examined the difference between two correlated, t distributed variables and aimed to derive a quantity with a distribution that would not depend on any population parameters. The math behind this derivation is too technical to be covered here, but in summation they used asymptotic expansion to find a function of the correlation between the variables that when used as a denominator to \((t_1 - t_2)\), where \(t_1\) and \(t_2\) are our t distributed variables, would approximate a t-distribution. The quantity found was:

\[\begin{equation} \psi=\frac{\frac{(x^*_a-\overline{x}_a)}{s_{a}}-\frac{(x^*_b-\overline{x}_b)}{s_{b}}}{ \sqrt{ (\frac{n+1}{n}) \left( (2-2 r)+ \frac{2(1-r^{2})}{n-1}+ \frac{(5+y^{2})(1-r^{2})}{2(n-1)^{2}}+ \frac{r(1+y^{2})(1-r^{2})}{2(n-1)^{2}}\right) }} \end{equation}\]

Where \(r\) is the correlation between the tasks and \(y\) the critical two-tailed t-value with \(n-1\) degrees of freedom. Garthwaite & Crawford (2004) demonstrate that \(\mathbb{P}[\psi > y]\approx\mathbb{P}[t >y]\), where \(\approx\) indicates approximate equivalence. To obtain a precise probability for \(\psi\) one solves for \(\psi = y\). See Garthwaite & Crawford (2004) and Crawford & Garthwaite (2005) for details. Choosing the positive root of \(\psi = y\) yields:

\[\begin{align} \begin{split} y & = \sqrt{\frac{ -b + \sqrt{b^2 - 4ac}}{2a}}, \text{where} \\ a & = (1+r)(1-r^2), \\ b & = (1-r)[4(n-1)^2+4(1+r)(n-1)+(1+r)(5+r)], \\ c & = - 2\left[\frac{X^*_{A} - \overline{X}_A}{s_A}-\frac{X^*_B -\overline{X}_B}{s_B}\right]^2\left(\frac{n(n-1)^2}{n+1}\right) \end{split} \tag{6} \end{align}\]

Where \(y\) is used as a t-statistic. This quantity is referred to as the revised standardised difference test (RSDT). Crawford & Garthwaite (2005) show with Monte Carlo simulations that this test is superior to both their own previous test (Crawford et al., 1998) and Payne & Gwynne Jones (1957)'s in controlling Type I errors. Even for very small sample sizes of \(n=5\), RSDT was shown to barely exceed the specified 5% error rate. So three valid frequentist tests remain, for deficits that is TD (3), for dissociations that is UDT (5) and RSDT (6).

For TD and UDT, Crawford & Garthwaite (2002) utilise the non-central t-distribution to set confidence limits on the abnormality of the case, something that grows ever more essential for reminding us that all test results come with uncertainty. However, it did not prove possible to set confidence limits on the estimations from the RSDT since it is only approximately t-distributed. This is one of the reasons why Crawford & Garthwaite (2007) wanted to develop a Bayesian method for estimating abnormality of discrepancy.

The test of deficit is most commonly used one-sided but the dissociation tests should in most cases be used two-sided as the direction of the effect solely depends on the order of the task scores.

There is one main difference between Bayesian and frequentist statistical
inference. In the Bayesian framework parameters (like means and standard
deviations) are treated as random variables with associated probability
distributions and if we believe a parameter has a certain distribution we update
that belief as more data is gathered and thus the parameter values can change.
Whilst, in the frequentist framework, parameters are treated as fixed attributes
of a population, estimations of which in a series or frequency (hence the
name) of trials will converge to the *true* value.

An intuitive way of
understanding the difference is to note how the two approaches phrase intervals
around parameters. The frequentist 95% *confidence interval* would cover the
population parameter 95% of the time. That is, if you estimate a population mean
from 100 different samples and create a confidence interval around each
estimation, 95 of these intervals would include the true population mean. The
Bayesian 95% *credible interval*, however, has a more intuitive interpretation.
For this interval you say that with a 95% certainty the population mean is
covered by the interval. That is because you take the values at the 2.5th and 97.5th
percentile of the parameter distribution to form the interval.

To estimate a parameter distribution Bayesians use prior knowledge of that parameter, i.e. they assign probabilities to possible values of the parameter depending on this knowledge — forming what is known as a prior distribution, or simply a prior. If no information exists one often apply a non-informative prior, the most simple of which assigns equal probabilities (uniform) to all possible parameter values. This is then updated when new information is obtained. The distribution formed by the updated prior is called the posterior distribution. The posterior probability of a hypothesis (i.e. a specified value of the parameter) is calculated by using Bayes theorem:

\[\begin{equation} \underbrace{\mathbb{P}[H \ |\ E]}_{\text{Posterior}} = \frac{\overbrace{\mathbb{P}[E \ | \ H]}^{\text{Likelihood}} \times \overbrace{\mathbb{P}[H]}^{\text{Prior}}}{\underbrace{\mathbb{P}[E]}_{\text{Marginal likelihood}}} \tag{7} \end{equation}\]

The \(H\) stands for hypothesis which may be affected by \(E\), evidence, i.e. the data (not used to estimate the prior). \(\mathbb{P}[E \ | \ H]\) is the probability of observing the data, given a hypothesis. \(\mathbb{P}[H \ |\ E]\) is the probability of a hypothesis given that some data have been observed. Say for example that we want to estimate IQ in a population with a sample of \(n=3\). By calculating the mean, standard deviation and standard error from this sample we can create a sampling distribution of the mean. The marginal likelihood, \(\mathbb{P}[E]\) will be a constant since it does not contain \(H\), we can therefore disregard that for now. If we assume a uniform prior, i.e. we do not believe that any hypothesis is more likely than another, \(\mathbb{P}[H]\) will also be a constant, reducing (7) to:

\[\begin{equation*} \mathbb{P}[H \ | \ E] = \mathbb{P}[E \ | \ H] \end{equation*}\]

Say that our sample had IQs of \(x_1 = 110, \ x_2= 115, \ x_3 = 120\). The sample
has a mean of 115, a standard deviation of 5 and a standard error of
\(5/\sqrt{n}\). Our theoretical sampling distribution of the mean (i.e. the
distribution of means if we draw a sample of \(n=3\) an infinite number of times)
would thus be a normal distribution with mean 115 and standard deviation =
standard error = \(5/\sqrt{n}\). To get the posterior distribution we now
calculate the probability of observing the data given another hypothesised mean.
To get the posterior probability for any
hypothesis, say \(\mu = 118\), we thus calculate:
\[\begin{align*}
\begin{split}
\mathbb{P}[H = 118 \ | \ E = (110, 115, 120)] = & \mathbb{P}[E = (110, 115, 120) | \ H = 118] =
\\ & \mathbb{P}[E = 110 \ | \ H = 118] \times \\ & \mathbb{P}[E = 115 \ | \ H = 118] \times \\
& \mathbb{P}[E = 120 \ | \ H = 118]
\end{split}
\end{align*}\]
That is, calculating the joint probability of observing \(x_1, \ x_2 \ \text{and} \ x_3\) given that our observations instead would have come from a distribution
with a mean of 118. This is then done for all possible hypotheses, i.e. values
of \(\mu\), creating a probability distribution of the parameter. The peak of this
distribution would in fact be the maximum likelihood estimate of the mean, which
is used in more classical estimation methods. Hence, using a non-informative prior
often yields estimations with frequentist properties^{2}.

However, if we for example have on good authority that mean IQ in the sampled population is 125 with an SD of 5, we specify our prior as such. This is then weighed in when calculating the posterior. That is, we calculate the joint probability that we have observed our data given a hypothesis (\(\mu\)) and the probability of observing that \(\mu\) in the prior distribution. This is done for all possible hypotheses, just as previously shown. Often one wants to use non-informative priors as to not arbitrarily bias the results. For an accessible and more thorough explanation of Bayesian statistics, see e.g. Donovan & Mickey (2019).

The example given here is a special case of Bayesian parameter estimation
that can be solved analytically.^{3} However, this is in many
cases not possible. Instead one utilise Markov Chain Monte Carlo (MCMC) methods.
A Markov Chain describe a series of events where the probability for each of
them solely depends on the one preceding it. Monte Carlo methods are mathematical
algorithms that solves problems by random generation of numbers. The idea
behind this in Bayesian statistics is that one estimates the posterior by drawing a large amount
of random samples from it. Thus “building” it from scratch. Because the denominator
in (7) is a constant and we can ignore it, this becomes possible.
Bayes theorem can then be rewritten as:
\[\begin{equation}
posterior \ \propto \ likelihood \times prior
\end{equation}\]
What this is saying is that the posterior density of a hypothesis
is proportional (\(\propto\)) to the likelihood of the data under
that hypothesis times the prior density of the hypothesis. The methods
for drawing these samples differ depending on the type of distribution
and problem at hand. But in general they are all building on algorithmic
rules (“recipes”) of drawing random numbers based on the likelihood and the prior,
saving them and after a large number of iterations observing the distribution
they form. The average of which often is the parameter of interest.

Assume a sample of \(n\) controls on which we measure some value x that is normally distributed with mean \(\mu\) and variance \(\sigma^2\). Let \(\overline{x}\) and \(s^2\) denote the sample mean and sample variance respectively. The case is denoted \(x^*\). The prior used is non-informative, see Crawford & Garthwaite (2007) and DeGroot & Schervish (2012) (p. 495) for the formal specification of the prior. The algorithm developed in Crawford & Garthwaite (2007) for obtaining a point estimate of a case’s abnormality \(p\), i.e. a p-value or the proportion of controls that would fall below the case and accompanying intervals is as follows:

Let \(\psi\) be a random draw from a \(\chi^2\)-distribution on \(n-1 \ df\). Then let \(\hat{\sigma}^2 = \frac{(n-1)s^2}{\psi}\) be the estimation of \(\sigma^2\) for this iteration.

Let \(z\) be a random draw from a standard normal distribution. Then let \(\hat{\mu}=\overline{x}+z\sqrt{(\hat{\sigma}^2/n)}\) be the estimate of \(\mu\) for this iteration.

With estimates of \(\mu\) and \(\sigma\), \(p\) is calculated conditional on these estimates being the correct \(\mu\) and \(\sigma\) by calculating \(z^*= \frac{x^* - \hat{\mu}}{\sqrt{\hat{\sigma}^2}}\). Let \(\hat{p}_i =\mathbb{P}[Z < z^*]\) be the estimate of \(p\) for this iteration. That is the probability of drawing a value less than \(z^*\) from a standard normal distribution.

Repeating these steps a large number of times will yield a distribution of \(\hat{p}\), the average of which is the point estimate of \(p\). If repeated e.g. 1000 times, the 25th smallest and 25th largest \(\hat{p}_i\) (\(i\) for iteration) is the lower and upper boundaries of the 95% Bayesian credible interval for \(p\).

Crawford & Garthwaite (2007) show that this method yields converging results to that of TD (3). As noted, this is often the case when using a non-informative prior, however, not always. For example, RSDT and its Bayesian analogue (BSDT) do not produce identical results. Crawford & Garthwaite (2007) showed that when there was no discrepancy between task A and B, but the case was severely impaired on both of them, RSDT exhibited an error rate much larger than the specified \(\alpha\)-level. For example, when the case had a deficit of 8 SD on both task A and B but no discrepancy, RSDT gave a false positive (Type I error) in \(34.72\)% of the simulations (\(n = 10\)). The BSDT on the other hand gave a false positive under the same circumstances in \(8.29\)% of the simulations. When there was no deficit on either task BSDT had a Type I error rate of \(7.32\)%. compared to RSDT with an error rate of \(4.6\)%. It is also argued that considering the often large effects of brain damage, this is an acceptable tradeoff.

Assume a sample of \(n\) controls on which we measure some value x and y from task A and B. Let \(\overline{x}\) and \(\overline{y}\) denote the sample means and \[\begin{equation*} \pmb{A}=\begin{pmatrix} s_{xx} & s_{xy} \\ s_{xy} & s_{yy} \end{pmatrix} \end{equation*}\] the sample covariance matrix (note that \(s_{xx}\) is not the same as \(s_{x}\) and denotes variance instead of standard deviation). It is assumed that the observations come from a bivariate normal distribution with mean \(\pmb{\mu}\) and variance \(\pmb{\Sigma}\), note that the parameters are bolded as to represent the vector and matrix:

\[\begin{equation*} \pmb{\mu} = \begin{pmatrix} \mu_x \\ \mu_y \end{pmatrix} \ \text{and} \ \pmb{\Sigma}=\begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{xy} & \sigma_{yy} \end{pmatrix} \end{equation*}\]

Let the case scores be denoted \(x^*\) and \(y^*\). Just as for the frequentist dissociation tests we want to estimate the proportion \(p\) of the control population that would exhibit a greater difference \(x-y\) than the case’s \(x^*-y^*\). A non-informative prior was again specified, see Crawford & Garthwaite (2007) and Jeffreys (1998). The algorithm for obtaining \(\hat{p}_i\), the \(i\)th estimation of \(p\) in Crawford & Garthwaite (2007) follows below:

Let \[\begin{equation*} \pmb{\widehat{\Sigma}}=\begin{pmatrix} \hat{\sigma}_{xx} & \hat{\sigma}_{xy} \\ \hat{\sigma}_{xy} & \hat{\sigma}_{yy} \end{pmatrix} \end{equation*}\] be a random draw from an inverse-Wishart distribution (a multivariate generalisation of the \(\chi^2\)-distribution) on \(n\) degrees of freedom with scale matrix \(\pmb{A}\). And let \(\pmb{\widehat{\Sigma}}\) be the estimate of \(\pmb{\Sigma}\) for this iteration.

Let \(z_1\) and \(z_2\) be two random draws from a standard normal distribution. Perform Cholesky decomposition on \(\pmb{\widehat{\Sigma}}\), that is finding the lower triangular matrix \(\pmb{T}\) such that \(\pmb{T}\pmb{T'}=\pmb{\widehat{\Sigma}}\). Then \[\begin{equation*} \pmb{\hat{\mu}} = \begin{pmatrix} \hat{\mu}_x \\ \hat{\mu}_y \end{pmatrix} = \begin{pmatrix} \overline{x} \\ \overline{y} \end{pmatrix}+ \pmb{T} \begin{pmatrix} z_1 \\ z_2 \end{pmatrix} \cdot \frac{1}{n} \end{equation*}\] is the estimation of \(\pmb{\mu}\) for this iteration.

With estimations of \(\pmb{\mu}\) and \(\pmb{\Sigma}\) we can calculate \(p\), given that they are the the correct values of \(\pmb{\mu}\) and \(\pmb{\Sigma}\). If an unstandardised test is desirable put: \[\begin{equation*} z^* = \frac{(x^* - \hat{\mu}_x) - (y^* - \hat{\mu}_y)}{\sqrt{\hat{\sigma}_{xx}+\hat{\sigma}_{yy}-2\hat{\sigma}_{xy}}} \end{equation*}\] If a standardised test is required, put: \[\begin{equation*} z_x = \frac{(x^* - \hat{\mu}_x)}{\sqrt{\hat{\sigma}_{xx}}}, \ z_y = \frac{(x^* - \hat{\mu}_y)}{\sqrt{\hat{\sigma}_{yy}}} \ \text{and} \ \hat{\rho}_{xy} = \frac{\hat{\sigma}_{xy}}{\sqrt{\hat{\sigma}_{xx}\hat{\sigma}_{yy}}} \end{equation*}\] and \[\begin{equation*} z^* = \frac{z_x - z_y}{\sqrt{2-2\hat{\rho}_{xy}}} \end{equation*}\]

Let \(\hat{p}_i\) be the tail area of a standard normal distribution less or greater than \(z^*\) (depending on alternative hypothesis). \(\hat{p}_i\) is then the estimate of \(p\) for this iteration. Repeating these steps a large number of times will yield a distribution of \(\hat{p}\), the average of which is the point estimate of \(p\). If repeated e.g. 1000 times, the 25th smallest and 25th largest \(\hat{p}_i\) is the lower and upper boundaries of the 95% Bayesian credible interval for \(p\).

The need to use matched samples when comparing a single case to a control sample has often often led to control samples being small. In an attempt to remedy this Crawford et al. (2011) followed up the previously described tests by developing methods that allow for the case to be assessed on abnormality in the presence of covariates. That is, the tests let you compare the case’s score on the task of interest conditioned upon the results of the controls having the same score on the covariate(s). If a patient has 15 years of education, his/her score on the task would be compared to the controls with equal length of education. This reduces the need to perfectly match control samples and is thus a major contribution to the field.

The procedural details of these tests are soon going to be updated in the
current vignette, but if used before that one thing is important to note. In
Crawford et al. (2011) they change the recommendation of the prior to use,
when testing for a dissociation. In Berger & Sun (2008) it is shown that
the prior used in Crawford & Garthwaite (2007) has good frequentist properties
for estimating \(\sigma_x\) and \(\sigma_y\) but less so for \(\rho\) or the
discrepancy size. Instead they recommend a calibrated prior. The main
difference being that an accept/reject algorithm is applied after the initial
draw from the inverse-Wishart distribution. Many Bayesians would argue that
good frequentist properties are not necessary when conducting a Bayesian
analysis. If so, one can use the “standard theory” Jeffreys (1998)'s
prior. In `singcar`

both types of priors are implemented for both tests. If
one wants to compare the results from them, the same prior should be used.

Berger, J. O., & Sun, D. (2008). Objective Priors for the Bivariate Normal Model. *The Annals of Statistics*, *36*(2), 963–982.

Crawford, J., & Garthwaite, P. (2002). Investigation of the single case in neuropsychology: Confidence limits on the abnormality of test scores and test score differences. *Neuropsychologia*, *40*(8), 1196–1208. https://doi.org/10.1016/S0028-3932(01)00224-X

Crawford, J., & Garthwaite, P. (2007). Comparison of a single case to a control or normative sample in neuropsychology: Development of a Bayesian approach. *Cognitive Neuropsychology*, *24*(4), 343–372. https://doi.org/10.1080/02643290701290146

Crawford, J., & Garthwaite, P. (2012). Single-case research in neuropsychology: A comparison of five forms of t-test for comparing a case to controls. *Cortex*, *48*(8), 1009–1016. https://doi.org/10.1016/j.cortex.2011.06.021

Crawford, J., & Garthwaite, P. (2005). Testing for Suspected Impairments and Dissociations in Single-Case Studies in Neuropsychology: Evaluation of Alternatives Using Monte Carlo Simulations and Revised Tests for Dissociations. *Neuropsychology*, *19*(3), 318–331. https://doi.org/10.1037/0894-4105.19.3.318

Crawford, J., Garthwaite, P., & Howell, D. (2009). On comparing a single case with a control sample: An alternative perspective. *Neuropsychologia*, *47*(13), 2690–2695. https://doi.org/10.1016/j.neuropsychologia.2009.04.011

Crawford, J., Garthwaite, P., Howell, D., & Gray, C. (2004). Inferential methods for comparing a single case with a control sample: Modified t‐tests versus mycroft et al.’s (2002) modified anova. *Cognitive Neuropsychology*, *21*(7), 750–755. https://doi.org/10.1080/02643290342000276

Crawford, J., Garthwaite, P., & Ryan, K. (2011). Comparing a single case to a control sample: Testing for neuropsychological deficits and dissociations in the presence of covariates. *Cortex*, *47*(10), 1166–1178. https://doi.org/10.1016/j.cortex.2011.02.017

Crawford, J., & Howell, D. (1998). Comparing an Individual’s Test Score Against Norms Derived from Small Samples. *The Clinical Neuropsychologist*, *12*(4), 482–486. https://doi.org/10.1076/clin.12.4.482.7241

Crawford, J., Howell, D., & Garthwaite, P. (1998). Payne and Jones Revisited: Estimating the Abnormality of Test Score Differences Using a Modified Paired Samples t Test. *Journal of Clinical and Experimental Neuropsychology*, *20*(6), 898–905. https://doi.org/10.1076/jcen.20.6.898.1112

DeGroot, M. H., & Schervish, M. J. (2012). *Probability and statistics* (4th ed). Addison-Wesley.

Donovan, T., & Mickey, R. M. (2019). *Bayesian Statistics for Beginners: A step-by-step approach* (1st ed.). Oxford University Press. https://doi.org/10.1093/oso/9780198841296.001.0001

Garthwaite, P., & Crawford, J. (2004). The distribution of the difference between two t -variates. *Biometrika*, *91*(4), 987–994.

Hassan, E. K., Sedda, A., Buckingham, G., & McIntosh, R. D. (2020). The size-weight illusion in visual form agnosic patient DF. *Neurocase*, 1–8. https://doi.org/10.1080/13554794.2020.1800748

Jeffreys, H. (1998). *Theory of probability* (3rd ed). Clarendon Press ; Oxford University Press.

Payne, R. W., & Gwynne Jones, H. (1957). Statistics for the investigation of individual cases. *Journal of Clinical Psychology*, *13*(2), 115–121.

Sokal, R. R., & Rohlf, F. J. (1981). *Biometry: The Principles and Practice of Statistics in Biological Research*. W. H. Freeman. https://books.google.co.uk/books?id=C-OTQgAACAAJ

In Crawford et al. (1998) they use the term ‘difference’ instead of discrepancy. This is a somewhat unfortunate usage since a difference can pretty much refer to anything. Hence, ‘discrepancy’ will be used for the most part when referring to the difference between scores from two tasks↩︎

A Bayesian 95% credible interval can for example be said to have good frequentist properties if it would cover the parameter 95% of the times it is created (as is the case for the frequentist 95% confidence interval)↩︎

The difference between analytic and numerical problem solving is that analytic solutions are exact as well as derived and presented in (for the mathematician) understandable forms. A numerical solution involves “guesswork” and is stopped when a solution is found that satisfies the problem. This method only approximates the “true” solution. Monte Carlo simulations belong to the latter category.↩︎