Conditional probability Statistics 1

Diagnostic tests A diagnostic test for a condition is said to be positive if it states that the condition is present and negative if it states that the condition is absent. Source: Agresti A. and C. Franklin (2009). Statistics: The Art of Science and Learning from Data.

Two types of error False positive: Test states the condition is present but it is actually absent False negative: Test states the condition is absent but it is actually present Source: Agresti A. and C. Franklin (2009). Statistics: The Art of Science and Learning from Data.

Down’s syndrome Down’s syndrome is a genetic disorder that was named after John Langdon Down The cause or causes of Down’s syndrome are unclear, but the single biggest risk factor for the condition seems to be the age at which a woman gives birth Source: http://www.nhs.uk/conditions/Downs-syndrome/Pages/Introduction.aspx [14 July 2011]

How common is it? Down’s syndrome is uncommon in the UK. For example, from 2007 to 2008, 1,843 cases of Down’s syndrome were diagnosed during pregnancy, and 743 babies were born with the condition. This means that about 1 in every 1,000 live births is affected by Down’s syndrome. Source: http://www.nhs.uk/conditions/Downs-syndrome/Pages/Introduction.aspx [14 July 2011]

Triple Blood Test A study of 5282 women aged 35 or over analysed the Triple Blood Test to test its accuracy. It was reported that of the 5282 women “48 of the 54 cases of Down’s syndrome would have been identified using the test and 25% of the unaffected pregnancies would have been identified as being at high risk”. Source: Agresti A. and C. Franklin (2009). Statistics: The Art of Science and Learning from Data.

Questions Estimate the probability of a positive test for a randomly chosen pregnant woman 35 years or older Given that the diagnostic result is positive, estimate the probability that Down’s syndrome is truly present Source: Agresti A. and C. Franklin (2009). Statistics: The Art of Science and Learning from Data.

Contingency table D...Down's syndrome Blood Test Total POS NEG D 48 6 54 D' 1307 3921 5228 1355 3927  5282

Sensitivity vs specificity Sensitivity measures the percentage of positives who are correctly identified as having the condition , that is Specificity measures the percentage of negatives who are correctly identified as not having the condition, that is

Random drug testing Air traffic controllers are required to undergo periodic random drug testing. A urine test used by the FAA has a sensitivity of 0.96 and a specificity of 0.93. Based on past drug testing, we can assume the probability of drug use is approximately 0.007 Source: Agresti A. and C. Franklin (2009). Statistics: The Art of Science and Learning from Data.

Questions A positive test result puts the air traffic controller’s job at risk. What is the probability of a positive test result? Find the probability an air traffic controller truly used drugs, given that the test is positive. Source: Agresti A. and C. Franklin (2009). Statistics: The Art of Science and Learning from Data.

Observations Most positive results occur with individuals who are not using drugs, since there are so many of such individuals (>99% of the population) In fact, the lower the prevalence (i.e. number of occurrences) ratio the lower the chance that a test result is truly correct! Source: Agresti A. and C. Franklin (2009). Statistics: The Art of Science and Learning from Data.