2.2 Diagnostic testing and screening

• •

  in some situations accuracy of classification may not be in question (such as death), however, most often the state of affairs will be less certain

• •

  it is thus of interest to quantify the tests’ reliability in the context of utility

• •

  typically assessment of a tests’ utility is achieved by applying the test to a number of individuals whose true disease status is known typically based upon a gold-standard

• •

  “the gold-standard is the most accurate existing test for the condition”

• •

  ‘gold-standards’ include histology in cancer diagnosis and microbiology for infectious diseases

• •

  disease classifications may be based on dichotomous test results (positive/negative or presence/absence) or based upon continuous measurements

• •

  for continuous measurements a cutoff level has to be determined

• •

  examples of continuous measures include:

  • –

    blood-sugar level for the diagnosis of diabetes

  • –

    cardiac enzyme levels for myocardial injury following myocardial infarction

  • –

    blood and brain changes in Alzheimer’s disease and dementia

• •

  two types of error can occur when testing: false positives and false negatives

• •

  sensitivity: the proportion of truly diseased persons in the tested population who are identified as diseased by the screening test probability of diagnosing a true case as diseased

  $$\text{sensitivity}=P(T\mid D)$$

• •

  specificity: the proportion of truly non-diseased persons who are so identified by the screening test (probability of diagnosing a truly non-diseased person as non-diseased):

  $$\text{specificity}=P(\overline{T}\mid \overline{D})$$

• •

  where $D$ and $\overline{D}$ denote diseased and non-diseased individuals, respectively, and $T$ and $\overline{T}$, respectively, denote positive and negative test results

• •

  also we may want to consider the probability that someone with a positive test result is actually diseased, and conversely, the probability that someone with a negative test result is actually disease free

• •

  positive predictive value (PPV) is the proportion of persons who are in fact diseased among those who test positive:

  $$\text{PPV}=P(D\mid T)$$

• •

  negative predictive value (NPV) is the proportion of persons who are in fact non-diseased among those who test negative:

  $$\text{NPV}=P(\overline{D}\mid \overline{T})$$

• •

  $\text{sensitivity}=\frac{a}{a+c}\times 100$

• •

  $\text{specificity}=\frac{d}{b+d}\times 100$

• •

  $\text{positive predictive value}=\frac{a}{a+b}\times 100$

• •

  $\text{negative predictive value}=\frac{d}{c+d}\times 100$

• •

  Note PPV and NPV are affected (potentially strongly) by the true prevalence of the disease

• •

  the values (as above) are typically multiplied by 100 and expressed as a percentage

• •

  when assessing tests based upon continuous measurements, the choice of the ’cut-off’ point is important, as this affects sensitivity and specificity

• •

  Example: Enzyme tests and myocardial infarction (MI): use of creatinine phosphokinase (CPK) assay in a coronary care unit. The data obtained were as follows:

  CPK activity	MI	non-MI
  0–49	2	32
  50–99	4	10
  100–149	6	5
  150–399	14	2
  400+	21	0
  Total no. patients	47	49

• •

  the four ‘sensible’ choices of cut-off point give sensitivities; specificities and PPV’s and NPV’s:

  Cutoff	Sensitivity	Specificity	PPV	NPV
  50	96	65	73	94
  100	87	86	85	88
  150	74	96	95	80
  400	45	100	100	65

• •

  it is clear from above that for this particular example:

  • –

    a low cut-off results in a more sensitive test;

  • –

    a high cut-off results in a more specific test;

  • –

    intermediate cut-offs result in smaller overall errors

• •

  choice of cut-off $\to$ strike a balance between high sensitivity and low specificity

• •

  each of the values in the previous table is calculated by forming a dichotomy based upon the test result: (+ve; -ve) for each cut-off point and then constructing a $2\times 2$ table in the form as previously. Below is the table for a cut-off point of CPK of 50 or greater.

• •

  sensitivity, specificity and their sum can be plotted to evaluate the various cut-points but typically a plot of sensitivity against (1 - specificity) is produced for the various cut-off values: the so-called receiver operating characteristic (ROC) curve

  Unnumbered Figure: Link

• •

  all things being equal one typically chooses the cut-off corresponding to the top-left-most point on this curve

• •

  prevalence, recall, is the proportion of the population of interest who have the disease at a given time point $\to$ contextually: essentially the prior probability of disease before observing the test result

• •

  assuming the data are a simple random sample from the general population of interest then an estimate is given by:

  $$\text{prevalence}=\frac{a+c}{N}$$

• •

  both PPV and NPV depend upon disease prevalence

• •

  if the study sample is not random (and this does not reflect the prevalence in the population of interest) then PPV and NPV do not yield a valid measure of the tests predictive value

• •

  sensitivity and specificity, however, are independent of the prevalence of disease

• •

  the positive predictive value $P(D\mid T)$ can be expressed using Bayes theorem:

• •

  if the sample is non-random the population prevalence $P(D)$ can be elicited from the population of interest and the predictive value of the test evaluated in context

• •

  the PPV equation above allows the prior probability of disease to be combined with the data regarding the test accuracy (i.e. the sensitivity and specificity) leading to a so-called posterior (post-test result) assessment of disease state given a positive test result

• •

  Example: a spinal fluid test is known to be 95% effective for detecting a disease when it is present and 99% effective when it is not. The prevalence of the disease is known to be 0.5% in the population of interest, thus we have

  $$\text{PPV}=\frac{(0.95)(0.005)}{(0.95)(0.005)+(1-0.99)(1-0.005)}$$

  $$\text{PPV}=0.33$$

• •

  and hence the post-test probability of disease for those testing positive is 33%

• •

  Note the NPV is given by:

  $$\text{NPV}=\frac{(0.99)(1-0.005)}{(0.99)(1-0.005)+(1-0.95)(0.005)}$$

  $$\text{NPV}=0.998$$

• •

  hence the post-test probability of being non-diseased for those testing negative is 99.8%

• •

  What if the disease prevalence was higher in the population?

• •

  therapeutic decisions with regard to tests are often considered using the likelihood ratio of a positive test result:

  $$\text{LR}=\frac{P(T\mid D)}{P(T\mid \overline{D})}=\frac{\text{sensitivity}}{1-\text{specificity}}$$

• •

  for the test to be informative we want the value to be high

• •

  the positive likelihood ratio is the ratio of the posterior disease odds (i.e. after witnessing a positive test result) to the prior disease odds and hence tells us how much the odds of disease increase given a positive test result

• •

  Example: A test has sensitivity of 90% and a specificity of 78% then

  $$\text{LR}=\frac{0.9}{0.22}=4.09$$

• •

  hence a positive test result implies you are 4 times more likely to have the disease

• •

  note that the LR may increase or decrease if the sensitivity and specificity of two competing diagnostic tests move in opposite directions

• •

  in general we need to consider the relative importance or weight that we will give to sensitivity

• •

  if the disease in question is life threatening and the treatment (subsequent to a positive diagnosis) has few side-effects then we would would want to give high weight to the sensitivity

• •

  if the disease in question is not too serious but treatment has many side-effects we would want to give more weight to specificity

• •

  thus if we give sensitivity weight $w$ and specificity $1-w$ we would want to maximise

  $$M=w\times \text{sensitivity}+(1-w)\times \text{specificity}$$

• •

  all things being equal we aim to maximise the number of correct decisions

• •

  to maximise with respect to the study (’sample criteria’) choose

  $$w=\frac{\text{number of positive samples}}{\text{total number of samples}}$$

• •

  to maximise with respect to the population criterion choose

  $$w=\frac{\text{number in population with the disease}}{\text{total population size}}$$

  which is the disease prevalence

• •

  based upon Youden’s criteria one chooses $w=0.5$

• •

  the choice of $w$ may be crucial in comparing tests

• •

  a screening test is a test for a particular disease given to populations at risk who are asymptomatic

• •

  the aim is to detect disease early with a view to better prognosis

• •

  screening tests are generally cheap (relatively) and typically followed up with more specific procedures to confirm diagnosis

• •

  screening tests are often based upon bio-markers from blood or urine samples

  • –

    PSA (prostate specific antigen) levels for prostate cancer

  • –

    carcinoembryonic antigen for colorectal cancer

• •

  clearly when screening for disease the sensitivity and specificity of the screening test is paramount

• •

  when screening the proportion who are truly diseased is likely to be small $\to$ many patients who are disease free will test positively and thus the acceptability of the test and follow-up need to be carefully considered

• •

  diagnostic testing in standard clinical practice is based upon patients presenting with disease symptoms $\to$ thus the weightings on test acceptability and test accuracy may differ

• •

  in order for a screening programme to be introduced several considerations need to be made (WHO)

• •

  the condition should be an important health problem

• •

  there should be a treatment for the condition

• •

  facilities for diagnosis and treatment should be available post screening

• •

  there should be a latent stage in the disease (early stage in the disease within which diagnosis is important)

• •

  there should exist a suitable screening test

• •

  the test should be acceptable to the population

• •

  the natural history of the disease should be well understood

• •

  there should be an agreed policy on who to treat

• •

  the total cost of finding a case should be economically balanced in relation to medical expenditure as a whole

• •

  case finding should be a continuous process not just a ’once and for all’ project

• •

  cancer screening:

  • –

    cervical cancer (pap smear)

  • –

    breast cancer (mammography)

  • –

    colorectal cancer (colonoscopy)

  • –

    melanoma (derma check)

  • –

    bowel cancer (faecal occult blood test)

• •

  foetal abnormalities

  • –

    alpha-fetoprotein

  • –

    blood tests

  • –

    ultrasound

In addition to test accuracy there are other issues to consider

• •

  lead-time bias: (survival time since diagnosis is longer with screening) $\to$ need to compare mortality in screened/non-screened groups

• •

  length-time bias: less severe cancers may be screen detected which may not be otherwise implicated prior to death so benefits seem greater

• •

  selection bias: sub-groups may be more likely to attend for screening such as those with family history

• •

  over-diagnosis abnormalities identified that would never cause a problem in the persons lifetime (e.g. prostate screening)

• •

  a randomised controlled trial (RCT) is recommended to demonstrate utility

• •

  Notwithstanding the above mentioned issues screening has resulted in better prognosis in many cases of disease. Consider, for example, the pap smear for cervical cancer.