Chi-square test is used to determine the statistically significant different between categorical variables. It also determines the difference between expected frequencies and the observed frequencies.

Mann Whitney U test is used to determine the statistically significant different between two independent groups.

Wilcoxon’s test is the test of dependency. it determines the statistically significant difference between two dependent groups.

Student t-test is one of the most commonly used method to test the hypothesis. It determines the significant difference between the means of two different groups.

ANOVA (analysis of variance) is similar to student’s t-test.

ANOVA is a statistical method used to determines the statistically significant difference between the mean of more than two group. In this experiment as we are dealing with three different group, ANOVA is most suitable test to determine the difference between each groups.

Mean, also known as the average, is the most common measure of central tendency. It is the sum of all observed values divided by the number of observation. It is not useful for skewed data, which has an abnormal distribution. It is useful, instead, for numerical data that have symmetric distribution. It reflects the contributions of each data in the group, and are sensitive to outliers.

The median is the value that falls in the middle position when the observations are ranked in order from the smallest to the largest. If the number of observations is odd, the median is the middle number. If it is even, the median is the average of the two middle numbers. Unlike the mean, the median is useful on skewed data, and can be used for ordinal or numerical data if skewed.

The mode is the value that occurs with the greatest frequency in a set of observations, and is utilized for bimodal distribution.

The variance and the standard deviation are not measures of central tendency, but of dispersion.

The data is ideal for chi square test evaluation as it will help determine if observed outcomes are in line with expected outcomes, and also if results are significant or due to chance.

The student’s t test is not ideal as it requires comparison of means from different populations, rather than proportions.

Pearson’s coefficient of linear regression is not ideal as it requires the plotting of a linear regression.

The numbers should be analysed before determining if there are any statistical conclusions that can be drawn from the population.

Statistical analysis is always required to determine the performance of any treatment during a clinical drug trial. Conclusions cannot be drawn simply by looking at the data.

Number needed to treat can be defined as the number of patients who need to be treated to prevent one additional bad outcome.

It can be found as:

NNT=1/Absolute Risk Reduction (rounded to the next integer since number of patients can be integer only).

Number needed to treat can be defined as the number of patients who need to be treated to prevent one additional bad outcome.

It can be found as:

NNT=1/Absolute Risk Reduction (rounded to the next integer since number of patients can be integer only).

where ARR= (Risk factor associated with the new drug group) — (Risk factor associated with the currently available drug)

So,

ARR= (0.04-0.03)

ARR= 0.01

NNT= 1/0.01

NNT=100

Number needed to harm are a measure of the impact of a treatment or intervention that is often used to communicate results to patients, clinicians, the public and policymakers. It states how many patients need to be treated for one additional patient to experience an adverse outcome (e.g. a death). It is calculated as the inverse of the absolute risk reduction. It can equally well be applied to harmful outcomes as well as beneficial ones, where it becomes numbers needed to treat (NNT) instead.

In this way, they are both calculated the same but NNT usually refers to a therapeutic treatment whereas NNH refers to a risk-factor for disease.

Attributable proportion is the proportion of disease that is caused due to exposure. It refers to the proportion of disease that would be eradicated from a particular population if the disease rate was diminished to match that of the unexposed group.

Risk ratio (relative risk) compares the probability of an event in an exposed (experimental) group to that of an event in the unexposed (control) group. Thus two are not the same.

The attributable risk is the rate of a disease in an exposed group to that of a group that has not been exposed to it i.e. how many deaths did the exposure cause.

Number needed to treat can be defined as the number of patients who need to be treated to prevent one additional bad outcome.

It can be found as:

NNT=1/Absolute Risk Reduction (rounded to the next integer since number of patients can be integer only).

where ARR= (Risk factor associated with the new drug group) — (Risk factor associated with the currently available drug)

So,

ARR= (165/1050)-(132/1044)

ARR= (0.157-0.126)

ARR= 0.031

NNT= 1/0.031

NNT=32.3

The relative risk (also known as risk ratio [RR]) is the ratio of risk of an event in one group (e.g., exposed group) versus the risk of the event in the other group (e.g., nonexposed group).

The odds ratio (OR) is the ratio of odds of an event in one group versus the odds of the event in the other group.

Its known that 68.3% of the total values of a normally distributed variable are found within a range of 1 standard deviation from the mean which makes the range to be 4.3 to 4.9 mmol/L.

A forest plot is a graphical representation that summarizes the findings of several research, such as a meta-analysis of a series of randomized controlled trials.

The Kaplan-Meier estimate shows survival over time, for example, plotting the number of patients still alive seven years after chemotherapy for lung cancer.

Fisher’s exact test similarly uses contingency tables to assess statistical significance, however, it is typically used when sample sizes are small.

Chi-square test assesses whether an association exists between two categorical variables using the observed and expected frequencies. For instance, is social class (I-V) related to body mass index (BMI) category? Using the observed and anticipated frequencies, the Chi-square test determines whether a connection exists between two categorical variables. For example, is socio-economic status related to BMI category?

Linear regression is a technique which attempts to model the relationship between two variables by fitting a linear equation to observed data. Linear regression uses correlation between two continuous variables. As correlation only indicates the strength of an association only, it cannot be used to forecast the change in one variable when a second variable is altered.

This equation takes the form y = mx + c, where ‘y’ is the dependent variable, ‘x’ is the independent variable, ‘m’ is the slope of the line and ‘c’ is the intercept. In this example, for a range of heights, it would be possible to map a line of best fit to a scatter plot and thus predict the forced expiratory volume (FEV1) for an individual.

9 & 27 can be obtained by subtracting and adding 9 from the mean. 9 is three times the standard deviation and we know that 99.7% values lie within 3 standard deviations from the mean. We can find the interval for 99.7% to verify in the following way:

For 99.7% confidence interval, you can find the range as follows:

1. Multiply the standard error by 3.

2. Subtract the answer from mean value to get the lower limit.

3. Add the answer obtained in step 1 from the mean value to get the upper limit.

4. The range turns out to be 9-27 kg.

The Delphi technique was developed in the 1950s and is a widely used and accepted method for achieving convergence of opinion concerning real-world knowledge solicited from experts within certain topic areas. Choosing the appropriate subjects is the most important step in the entire process because it directly relates to the quality of the results generated, despite this, there is no exact criterion currently listed in the literature concerning the selection of Delphi participants.

Therefore, due to the potential scarcity of qualified participants and the relatively small number of subjects used in a Delphi study, the ability to achieve and maintain an ideal response rate can either ensure or jeopardize the validity of a Delphi study.

The question mentions a quantitative, ratio scale data set. The use of mean would be ideal under normal circumstances, however, in this situation median is preferred as it is less sensitive to the skewness of data. The median is usually preferred to other measures of central tendency when your data set is skewed (i.e., forms a skewed distribution)

The level of statistical significance required influences the sample size used. This is because sample size is used in the calculation of SD/SE.

Sample size does not affect

The level of acceptance
The alternative hypothesis with a general level set at p<0.05
The test to be used.

Experience of the investigator and the type of patient recruited should have no bearing on the required sample size.

This clinical trial consists over 1000 patients being evaluated for the response to a new treatment against a currently licensed treatment for ulcerative colitis. Therefore, it is comparing its efficacy to an established therapeutic or control in a larger population of volunteers. These are the characteristics of a phase III clinical trial.

Work up bias involves comparing the novel diagnostic test with the current standard test. A portion of the patients undergo the standard test while others undergo the new test as the standard test is costly. The result can be alteration in specify and sensitivity.

Selection bias is when randomisation is not achieved.

Attention bias refers to the person’s failure to consider various alternatives when he pre occupied by some other thoughts.

Instrument bias is related to the experience and extent of familiarization of the participating individuals with the test.

Co intervention bias is characterized by the groups receiving different co interventions.

Phase 0 trials assist the scientists in studying the behaviour of drugs in humans by micro dosing patients. They are used to speed up the developmental process. They have no measurable therapeutic effect and efficiency.

Tests are used to confirm the presence of a particular disease. However the results can be misleading at times since most of the tests have some limitations associated with them.
Sensitivity is the correct term that refers to the probability of a positive test. The others are explained below:

False Positive rates refer to the proportion of the patients who don’t have the condition being detected as positive.

False Negative rates refer to the proportion of the patients who have the condition being detected as negative (like the 20% of the patients that went undetected by the Ca-125 test).

Specificity describes the ability of a test to detect and pick up people without the disease. Absolute risk ratio compares the rate of two separate outcomes.

Work up bias involves comparing the novel diagnostic test with the current standard test. A portion of the patients undergo the standard test while others undergo the new test as the standard test is costly. The result can be alteration in specify and sensitivity.

Recall bias is specifically appropriate to the case control studies that is when ever the memories retrieved by the participants differ in accuracy.

Not publishing the results of a valid study just because they are negative or uninteresting can be termed as publication bias.

When information gathering is ill suited with respect to time i.e. collecting the data regarding a fatal disease many years after the death of its patients, it is termed as Late – look bias.

The case in point is an instance of lead time bias when upon comparing two tests, one is able to detect the condition earlier than the other but the overall outcome doesn’t change. There is a possibility that this will make the survival rates for the newer test look more promising.

Randomisation allows for performance of experimental trials in a random order. Using this method gives us control over the confounding variables that are not supposed to be held constant.

For an instance, by employing randomisation we get to control biological differences among individual human beings during experimental trials.

The Weibull distribution are used to describe various types of observed failures of the components. it is used in reliability and survival analysis.

Regression Analysis is used to measure the relationship between among two or more variable. It determines the effect of independent variables on the dependent variables.

Student t-test is one of the most commonly used method to test the hypothesis. It determines the significant difference between the means of two different groups.

A time series is a collection of observations of well-defined data obtained at regular interval of time.

Kaplan-Meier estimator is used to estimate the survival function from lifetime data. It can be derived from maximum likelihood estimation of hazard function. It is most likely used to measure the fraction of patient’s life for a certain amount of time after treatment.

Boyle’s law is correctly stated as it states that the volume of a gas of known mass is inversely proportional with absolute pressure, at a constant temperature.

Beer’s law states that radiation absorption by a solution of known thickness and concentration is identical to that of a solution of double thickness and half concentration.

Bougner’s (or Lambert’s) law states that every layer with the same thickness will absorb the same amount of radiation as it passes through.

Graham’s law states that the diffusion rate of a gas is inversely related to the square root of its molecular weight.

Raoult’s law states that the reduction of a solvent’s vapour pressure is directly proportional to the solute’s molar concentration.

The attributable risk is the rate of a disease in an exposed group to that of a group that has not been exposed to it. It involves the measure of association that is pertinent to making decisions for the individuals.

Observation bias occurs when the behaviour of an individual changes that results from their awareness of being observed.

Recall bias introduced when participants in a study are systematically more or less likely to recall and relate information on exposure depending on their outcome status.

Attrition bias is a systematic error caused by unequal loss of participants from a randomized controlled trial (RCT). In clinical trials, participants might dropout due to unsatisfactory treatment or efficacy, intolerable adverse events, or even death.

Selection bias introduced when the individuals are not chosen randomly to take a part in the study. It usually occurs when the research decides who is going to be studied, they are not the representative of the population.

Lead-time bias occurs when a disease is detected by a screening test at an earlier time point rather than it would have been diagnosed by its clinical appearance. In this bias, earlier detection improves the survival time in the intervention group.

Funnel plot is essentially a scatterplot of the effect of treatment against a particular measure of study precision. Its primal purpose is to serve as a visual aid and help in detection of bias or systematic heterogenity.

Half life (T½) is the time required to change the amount of drug in the body by one-half (or 50%) during elimination. The time course of a drug in the body will depend on both the volume of distribution and the clearance.

Extrapolating the values from the plasma concentration vs time:

Plasma concentration at 0 hours = 800 mcg/mL
Plasma concentration at 2 hours = 400 mcg/mL
Plasma concentration at 4 hours = 200 mcg/mL
Plasma concentration at 6 hours = 100 mcg/mL
Plasma concentration at 8 hours = 50 mcg/mL
Plasma concentration at 10 hours = 25 mcg/mL
Plasma concentration at 12 hours = 12.5 mcg/mL
Plasma concentration at 14 hours = 6.25 mcg/mL

Correlation doesn’t justify causality. Correlation coefficient gives us an idea whether or not the two parameters provide have any relation of some sort or not i.e. does change in one prompt any change in other? It has nothing to do with predictions. For that purpose linear regression is used.

Recall bias introduced when participants in a study are systematically more or less likely to recall and relate information on exposure depending on their outcome status.

In procedure bias, the researcher decides assignment of a treatment versus control and assigns particular patients to one group or the other non-randomly. This is unlikely to have occurred in this case, although it is not mentioned specifically.

Self Selection or volunteer bias occur when those subjects are selected to participate in the study who are not the representative of the entire target population. those subjects may be from high socio-economic status and practice those activities or lifestyle that improves their health.

Lead-time bias occurs when a disease is detected by a screening test at an earlier time point rather than it would have been diagnosed by its clinical appearance. In this bias, earlier detection improves the survival time in the intervention group.

Variance and standard deviation indicate the dispersion of the plot from mean value and thus are not really helpful in summarizing the mean.

Range is the difference between maximum and minimum value that is 45 in this case.

The mean in this case is 6.2 due to the presence of an outlier 45. In the presence of outlier mean can be misleading as it is quite sensitive to skewness in data.

Mode is the most frequent value. In this case mode has 4 values: 0,1,2,4.

In case of skewedness, median is the most apt representative of the mean as it is not affected by outliers. In this case since the data set has even values i.e. 10. Median is the average of the 5th & 6th entry after arranging the data in ascending order like that in case of the question (0,0,1,1,2,2,3,4,4,45). This turns out to be 2.