The row totals are the total number of patients receiving Drug A and placebo, whereas the column totals are the total number of patients who died and survived. The chi-square test can be used to determine if there is a statistically significant association between death and treatment with Drug A. The H 0 would be there is no association between death and treatment with Drug A, and the H A would be there is an association between death and treatment with Drug A. Using the chi-square test, p < 0.001 therefore rejecting the H 0, and there is an association between death (or improved survival) with Drug A. Of note, the Fisher’s exact test is used when the expected cell frequencies are <5. The expected cell frequency is the probability of being in a given cell times the total sample size. For example, the expected cell frequency for the upper left cell is calculated as (525 × 102)/1,100 = 48.7.
4.3.3 Relative Risk and Odds Ratio
The relative risk (RR) is the ratio of the incidence in people with the risk factor (exposed persons) to the incidence in people without the risk factor (nonexposed persons). RR can only be calculated for cohort studies and clinical trials. In both instances, there is a group of subjects with the risk factor and a group of subjects without the risk factor. The subjects are then followed over time to determine which subjects develop the outcome of interest.
The odds ratio (OR) is the odds that a subject with an adverse event was at risk divided by the odds that a subject without an adverse event was at risk. OR can be calculated for cohort and case–control studies. The OR and RR can be easily calculated using a 2 × 2 table.
OR = a * d/b * c
For example, a trial was performed comparing thrombotic events in patients taking a nonsteroidal anti-inflammatory drug (NSAID) compared to placebo. In the NSAID group, 46 out of 1,000 patients had a thrombotic event compared to 26 out of 1,000 patients in the placebo group.
The calculated OR would be 1.81 [(46 × 974)/(954 × 26)]. This can be interpreted as patients taking the assigned NSAID have 1.81 increased odds of having a thrombotic event compared to patients taking placebo. The calculated RR [(46/1,000)/(26/1,000)] is 1.76. This can be interpreted as patients taking the assigned NSAID have a 76 % increase in the rate of thrombotic events compared to patients taking placebo.
OR or RR greater than 1 indicates that there is an increased risk of the measured event associated with the exposure. When the OR or RR equals 1, the measured event is no more likely to occur with or without the exposure. On the other hand, when the OR or RR is less than 1, the measured event is less likely to occur with the exposure . Also of note, in the previous example, OR and RR approximate each other, 1.81 and 1.76, respectively. This is usually true when the event rates are low and/or the treatment effect is small.
Other terms to be familiar with include absolute risk reduction, number needed to treat, absolute risk increase, and relative risk reduction. The absolute risk reduction allows one to assess the reduction in risk compared with the baseline risk. Specifically, it is the reduction in risk of a new intervention compared to the risk without intervention, and it is the absolute value of the difference between the experimental and control event rates. The number needed to treat is the reciprocal of the absolute risk reduction and provides the number needed to treat in order to prevent one event. For example, if a new treatment decreases the relative risk of myocardial infarction and has an absolute risk reduction of 0.0086, then the number of people who need to be treated to prevent 1 myocardial infarction is approximately 116 (1/0.0086 = 116.3). Absolute risk increase is the opposite of the absolute risk reduction. It is the increase in risk with a new treatment compared with the risk without the treatment, and relative risk reduction is the reduction in risk with a new treatment relative to the risk without treatment .
4.3.4 Correlation and Linear Regression
Correlation is used to determine if a linear relationship exists between two quantitative variables. Linear correlation is a measure of the degree to which an increase or decrease in one continuous variable is associated with a proportional increase or decrease in a second continuous variable . In other words, can the relationship between two variables be described by a straight line? For example, consider a scatterplot depicting the hemoglobin A1c and serum glucose in ten patients with diabetes mellitus. If every point falls on a straight line, the two variables are perfectly correlated. The Pearson correlation coefficient (r) can be used to calculate the strength of a relationship and ranges from −1 to +1. A value of 0 represents no correlation, −1 represents perfect negative correlation, and +1 represents perfect positive correlation between two variables. The Pearson’s correlation coefficient can be calculated for any dataset, but it is more meaningful if the two variables are normally distributed.
Linear regression allows investigators to analyze the relationship between two or more continuous variables when one variable depends on the others and allows investigators to predict one variable given the value of the other variables . For example, investigators were interested in the relationship between height and forced expiratory volume (FEV) in children. Using linear regression, it was found that FEV = −6.07 + (0.14 × height). Using this equation, the predicted FEV for a five foot (60″) child would be 2.34 l (−6.06 + 0.14 × 60). Of note, when performing multivariate analysis (i.e., more than two variables are included in the model), the number of covariates used in the model depends on the sample size. Ideally the sample size should exceed ten times the number of independent variables. For example, if the sample size in a study is 100, no more than ten independent variables should be included in the linear regression model. If too many independent variables are included in the model, investigators run the risk of overfitting the data. The same is also true for small sample size. Also, assumptions must be met in order to utilize linear regression models. They are as follows: the sample must be randomly selected, X and Y are normally distributed, and the Y values are independent of each other (i.e., not correlated).
4.3.5 Survival Analysis
Survival analysis is also referred to as time to event analysis. It allows for the analysis of binary categorical outcomes such as death, onset of disease, recurrence of disease, and onset of disability. Survival can be reported as a percentage (i.e., 1-year or 5-year survival), median survival, or survival curves. There are several different survival analysis methods: incidence density method, life table (actuarial method), Kaplan-Meier (product-limit method), and Cox Proportional Hazards model. Kaplan-Meier and Cox Proportional Hazards are the more commonly used survival methods for clinical trials. Kaplan-Meier survival analysis allows investigators to generate survival curves for each group which can be compared using the logrank statistic. Kaplan-Meier survival analysis is to be considered generally reliable up to two times the median follow-up time. Assumptions made when utilizing the Kaplan-Meier method include no change in the event rate over time and the outcome is the same for patients that are followed and those lost to follow-up. The Cox Proportional Hazard model provides a hazard ratio and allows for the comparison of two or more survival curves after adjusting for covariates. For example, a multi-institutional retrospective study identified 3,500 patients who underwent pancreatic resection for pancreatic cancer. A multivariate-adjusted Cox Proportional Hazard model was used to evaluate the prognostic significance of adjuvant radiation therapy (AXRT). The hazard ratio for patients who received AXRT was 0.75. This can be interpreted as patient who received AXRT after surgical resection of pancreatic cancer had a 25 % decreased risk of death compared to patients who did not receive AXRT. During covariate adjustment, in general the ratio of the number of independent variables used in the Cox model to number of events should not exceed 1:10. For example, if a study has a sample size of 1,000 and 100 patients died, the maximum number of independent variables that should be included in the model is 10. Similar to linear regression, if too many independent variables are included in the model, one is at risk of overfitting the data. Assumptions for Cox regression are the same as Kaplan-Meier survival analysis, and the effect of the covariate does not change over time for any of the independent variables.
4.4 Analyzing the Data
Careful analysis of data obtained from clinical trials requires a major investment of time and effort. Inappropriate statistical analysis can result in misleading conclusions and impairs the credibility of the trial and investigators. Two important issues that should be considered in the analysis of clinical trial results are intention-to-treat analysis and the role for subgroup analysis.
4.4.1 Intention-to-Treat Analysis
Intention-to-treat (ITT) analysis is a technique commonly used in randomized control trials. The definition is as follows: “All patients randomly allocated to one of the treatments in a trial should be analyzed together as representing that treatment, whether or not they completed, or indeed received that treatment” . In other words, ITT compares outcomes between study groups with each participant analyzed according to their randomized group assignment regardless of receiving the assigned treatment, withdrawal from the study, or deviation from the protocol.
An alternative to ITT is “per protocol” analysis, which only evaluates those participants who complied with the assigned treatment. This appears to be an appropriate approach to analysis because participants can only be affected by an intervention they actually received. However, the problem arises when participants who adhere to the study treatment differ from those that are noncompliant or drop out, thus introducing bias . For example, in the Postmenopausal Estrogen/Progestin Intervention (PEPI) Trial, 875 healthy postmenopausal women aged 45–64 years of age who had no known contraindication to hormone therapy were randomly assigned to four different estrogen or estrogen plus progesterone regimens and placebo. Of the 175 women assigned to the unopposed estrogen arm, 41 (23 %) discontinued treatment because of endometrial hyperplasia, which is a precursor of endometrial cancer . If “per protocol” analysis was performed, these women would have been eliminated from analysis, and the association of estrogen therapy and endometrial cancer may have been missed.
ITT analysis not only minimizes bias but it also maintains the similarities between treatment groups in regard to prognosis. This is the reason for randomization, and this feature may be lost if analysis is not performed on the groups produced by the randomization process. This can be illustrated by the European Coronary Surgery Study Group Trial comparing medical and surgical treatment for stable angina. A total of 768 men under the age of 65 with angina were included in the study (373 men were randomized to medical treatment, and 395 men were randomized to surgical treatment). A total of 26 men assigned to the surgical arm did not undergo surgery, and 50 men assigned to the medical arm underwent surgery. Using ITT analysis, there was no significant difference in mortality between the two groups at 2 years . Alternatively, using “per protocol” analysis, the mortality rate would be 8.4 % for the medical treatment and 4.1 % for the surgical treatment (P = 0.018) . In “per protocol” analysis, surgery appears to have a falsely low mortality rate.