Testing for a deficit

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 
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls

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 
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient 
CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls

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 
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls

# Extracting the coviariate below
DF_age <- size_weight_illusion[size_weight_illusion$PPT == "DF", "YRS"] # Patient
CON_age <- size_weight_illusion[size_weight_illusion$PPT != "DF", "YRS"] # Controls

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

Testing for a dissociation

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 
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls

# Extracting scores from the kinesthetic size-weight illusion from size_weight_illusion 
DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient
CON_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls

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 
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls

# Extracting scores from the kinesthetic size-weight illusion from size_weight_illusion 
DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient
CON_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls

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 
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls

# Extracting scores from the kinesthetic size-weight illusion from size_weight_illusion 
DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient
CON_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls

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 
DF_V_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "V_SWI"] # Patient
CON_V_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "V_SWI"] # Controls

DF_K_SWI <- size_weight_illusion[size_weight_illusion$PPT == "DF", "K_SWI"] # Patient
CON_K_SWI <- size_weight_illusion[size_weight_illusion$PPT != "DF", "K_SWI"] # Controls

# Extracting the coviariate below
DF_age <- size_weight_illusion[size_weight_illusion$PPT == "DF", "YRS"] # Patient
CON_age <- size_weight_illusion[size_weight_illusion$PPT != "DF", "YRS"] # Controls

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.

Power calculators

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

Frequentist approaches

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.

Bayesian approaches

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 properties².

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.