# Comparing a single (patient) score to controls

Crawford and Howell (1998) propose a test that compares a single patient score to those obtained from a control group. This is a more conservative test that the usual one sample t-test because it assumes the patient score has a variance equal to that of the controls rather than treating this score as a constant with zero variance.

This test may be implemented using this spreadsheet for raw data or Crawford's own program for summary data. See also here.

An alternatively similar approach used in published imaging studies compares scores from a group of controls with those of individual patients using the same tests. For example, suppose we wish to obtain standardised scores to see how impaired a patient's Culture Fair score is compared to that predicted using the their NART score assuming the NART had come from a control group. This can be achieved by, firstly, obtaining the intercept and regression estimate for NART from a simple regression of NART on Culture Fair from a group of controls. These two coefficients can then be used to obtain the predicted score of the patient using the *patient's own* NART score. The standardised residual score for this patient, z(patient), is then equal to

$$\mbox{z(patient)} = \frac{\mbox{Observed Patient Culture Fair Score - (control intercept + control NART regression coefficient x patient NART)}}{\mbox{sd(residuals in controls)}}

where sd(residuals in controls) is the residual standard deviation in the controls equal to the square root of the *Mean Square of the residual* outputted in the ANOVA table from the simple regression using NART as a predictor of Culture Fair using the controls.

Z-scores calculated as above can also be combined by averaging or summation for different outcome measures looking at aspects (e.g. Culture Fair, AH4) of the same underlying measure (e.g. general intelligence) observed on both patients and controls. Such z-scores correspond to the definition of standardised residuals evaluated in, for example, SPSS when performing simple and multiple linear regressions.