The sensitivity of a screening test can be described in a variety of ways, typically such as sensitivity being the ability of a screening test to detect a true positive, being based on the true positive rate, reflecting a test’s ability to correctly identify all people who have a condition, or, if 100%, identifying all people with a condition of interest by those people testing positive on the test.The specificity of a test is defined in a variety of ways, typically such as specificity is the ability of a screening test to detect a true negative, being based on the true negative rate, correctly identifying people who do not have a condition, or, if 100%, identifying all patients who do not have the condition of interest by those people testing negative on the test.Sensitivity=[a/(a+c)]×100Specificity=[d/(b+d)]×100a: True positiveb: False Positivec: False negatived: True negative

Before the test, the patient was likely to have the disease being tested for. Their pre test odds were 2, making them twice as likely to have the disease. After testing positive, their pretest odds of 2 and the likelihood ratio of 3 are multiplied to produce 6 as the post test odds. The patient is now 6 times as likely to have the disease after the test. Likelihood ratios above 10 or below 0.1 are strong evidence to rule in and rule out a diagnosis. The negative likelihood ratio is not known here, but does not subtract from the interpretation of the outcomes in this patient.

Crossover trials are characterized by the switching of study participants throughout the treatment groups, keeping a random order of switching. This randomization is important to determine any carry-over effect of different treatment modalities. The benefit of this study is the evaluation of the efficacy of various short-term treatment options for the relief of chronic conditions. The washout period is small in this type of study. Crossover studies are more efficient than the parallel studies, but they should be used according to the treatment options and outcomes.

Case control studies are used to determine the relationship between exposure to a risk factor and the resultant outcomes. It finds patients in the general population who have a certain condition and retrospectively searches for past exposure to possible risk factors for the disease. Controls are people who do not have the disease found in the general population. This type of study is useful for rare diseases, and is less time consuming to conduct. While cohort studies can be used to study rare diseases they measure the relative risk of developing the disease over time based on exposure. Ecological or cross-sectional studies attempt take a snapshot of a whole population, and thus are inappropriate for rare diseases as larger sample numbers are needed. A randomised control trial is better suited to determine the effect of an intervention.

To quantify differences between percentages you can use Fisher’s exact test.Odds ratios, relative risks and number needed to treat are ways of quantifying differences between percentages in two groups, however are not in themselves significance tests.

In the final result of this double blind study, 25% or one in four of patients who received the treatment responded positively. Though the difference is significant at P<0.05, it is better to present this as a significant difference at 0.4%. The number needed to treat (NNT) is defined as the inverse of the absolute risk reduction. Taking this as 0.25 (25%), the NNT is 1/0.25 = 4. However given the 95% confidence interval is between 10-50% for absolute risk reduction, the NNT with 95% CI is between 2 and 10. Therefore we cannot say with certainty that the NNT is at least 4.

The Chi-squared test evaluates if two variables are related. The other statistical tests mentioned do not perform this function.

The type of significance test used depends on whether the data is parametric (can be measured, usually normally distributed) or non-parametric.
Parametric tests:
Student’s t-test – paired or unpaired*
Pearson’s product-moment coefficient – correlation

Non-parametric tests:
Mann-Whitney U test – unpaired data
Wilcoxon signed-rank test – compares two sets of observations on a single sample
chi-squared test – used to compare proportions or percentages
Spearman, Kendall rank – correlation.

The outcome measured is not normally distributed, i.e. it is non-parametric. This excludes the Student’s t-tests. We are not comparing percentages/proportions so the chi-squared test is excluded. The Mann–Whitney U test is a nonparametric test of the null hypothesis that it is equally likely that a randomly selected value from one sample will be less than or greater than a randomly selected value from a second sample.
This test can be used to investigate whether two independent samples were selected from populations having the same distribution.

While the results of this before and after study can provide some preliminary insight into the effects of the dietary intervention, the results should be viewed with great caution as they may be considered merely anecdotal. The lack of a control group contributes to confounding the data. Therefore, while the difference may be statistically significant and the confidence interval is compatible with a fall in the number of seizures after therapy, the results cannot be introduced as the standard without a control group to compare with.

Relative risk is the probability of an outcome occurring in an exposed group as compared to the probability of that outcome in an unexposed group. In the scenario given, 50% of the children exposed developed the disease while only 10% of the children who were unexposed developed the disease. The relative risk was therefore 50/10=5. There is no further information about whether the study was controlled, the confidence interval, or the type of study. It therefore cannot be confirmed whether the chemical is causative and if it should be avoided.