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##### Clinical Statistics for Non-Statisticians – Part II

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**Clinical Statistics for Non-Statisticians – Part II**Kay M. Larholt, Sc.D. Vice President, Biometrics & Clinical Operations Abt Bio-Pharma Solutions**Topics**• Review of Statistical Concepts • Hypothesis Testing • Power and Sample Size • Interim Analysis**Statistics**Per the American Heritage dictionary - “The mathematics of the collection, organization, and interpretation of numerical data, especially the analysis of population characteristics by inference from sampling.” • Two broad areas • Descriptive – Science of summarizing data • Inferential – Science of interpreting data in order to make estimates, hypothesis testing, predictions, or decisions from the sample to target population.**Introduction to Clinical Statistics**• Statistics - The science of making decisions in the face of uncertainty • Probability - The mathematics of uncertainty • The probability of an event is a measure of how likely the event is to happen**Measures of central tendency**Mean, Median, Mode Measures of dispersion Range, Variance, Standard deviation Measures of relative standing Lower quartile (Q1) Upper quartile (Q3) Interquartile range (IQR) range (IQR) Descriptive Statistics for Continuous Variables**Basic Probability Concepts**Sample spaces and events Simple probability Joint probability**Probability**1 Certain • Probability is the numerical measure of the likelihood that an event will occur • Value is between 0 and 1 .5 0 Impossible**Number of event outcomes**P( E ) = Total number of possible outcomes in the sample space Computing Probabilities The probability of an event E: Assumes each of the outcomes in the sample space is equally likely to occur**Gaussian or Normal Distribution aka “Bell Curve”**• Most important probability distribution in the statistical analysis of experimental data. • Data from many different types of processes follow a “normal” distribution: • Heights of American women • Returns from a diversified asset portfolio • Even when the data do not follow a normal distribution, the normal distribution provides a good approximation**Gaussian or Normal Distribution aka “Bell Curve”**The Normal Distribution is specified by two parameters • The mean, • The standard deviation, **m=0**Standard Normal Distribution =1**Characteristics of the Standard Normal Distribution**• Mean µ of 0 and standard deviation σof 1. • It is symmetric about 0(the mean, median and the mode are the same). • The total area under the curve is equal to one. One half of the total area under the curve is on either side of zero.**Area in the Tails of Distribution**• The total area under the curve that is more than 1.96 units away from zero is equal to 5%. Because the curve is symmetrical, there is 2.5% in each tail.**Normal Distribution**• 68% of observations lie within ± 1 std dev of mean • 95% of observations lie within ± 2 std dev of mean • 99% of observations lie within ± 3 std dev of mean**Sample versus Population**• A population is a whole, and a sample is a fraction of the whole. • A population is a collection of all the elements we are studying and about which we are trying to draw conclusions. • A sample is a collection of some, but not all, of the elements of the population**Sample versus Population**• To make generalizations from a sample, it needs to be representative of the larger population from which it is taken. • In the ideal scientific world, the individuals for the sample would be randomly selected. This requires that each member of the population has an equal chance of being selected each time a selection is made.**Randomisation**• To guard against any use of judgement or systematic arrangements i.e to avoid bias • To provide a basis for the standard methods of statistical analysis such as significance tests • Assures that treatment groups are balanced (on average) in all regards. • i.e. balance occurs for known prognostic variables and for unknown or unrecorded variables**Inferential statistics calculated from a clinical trial make**an allowance for differences between patients and that this allowance will be correct on average if randomisation has been employed.**Hypothesis Testing**• Steps in hypothesis testing: state problem, define endpoint, formulating hypothesis, - choice of statistical test, decision rule, calculation, decision, and interpretation • Statistical significance: types of errors, p-value, one-tail vs. two-tail tests, confidence intervals**Descriptive and inferential statistics**• Descriptive statistics is devoted to the summarization and description of data (population or sample) . • Inferential statistics uses sample data to make an inference about a population .**Objectives and Hypotheses**• Objectives are questions that the trial was designed to answer • Hypotheses are more specific than objectives and are amenable to explicit statistical evaluation**Examples of Objectives**• To determine the efficacy and safety of Product ABC in diabetic patients • To evaluate the efficacy of Product DEF in the prevention of disease XYZ • To demonstrate that images acquired with product GHI are comparable to images acquired with product JKL for the diagnosis of cancer**How do you measure the objectives?**• Endpoints need to be defined in order to measure the objectives of a study.**Endpoints: Examples:**• Primary Effectiveness Endpoint – • Percentage of patients requiring intervention due to pain, where an intervention is defined as : • Change in pain medication • Early device removal**Endpoints: Examples:**• Primary Endpoint: Percentage of patients with a reduction in pain: • Reduction in the Brief Pain Inventory (BPI) worst pain scores of ≥ 2 points at 4 weeks over baseline.**Endpoints: Examples**• Patient Survival • Proportion of patients surviving two years post-treatment • Average length of survival of patients post-treatment**Objectives and Hypotheses**• Primary outcome measure • greatest importance in the study • used for sample size • More than one primary outcome measure - multiplicity issues**Hypothesis Testing**• Null Hypothesis (H0) • Status Quo • Usually Hypothesis of no difference • Hypothesis to be questioned/disproved • Alternate Hypothesis (HA) • Ultimate goal • Usually Hypothesis of difference • Hypothesis of interest**Decision Making**“Truth” Decision Type I Error**Decision Making**“Truth” Decision Type I Error**Decision Making**“Truth” Decision Type I Error**Decision Making**“Truth” Test Type I Error**Decision Making**“Truth” Decision Type I Error**Hypothesis Testing**Type I Error – Society’s Risk Type II Error – Sponsor’s Risk**The Type I Erroroccurs when we conclude from an experiment**that a difference between groups exists when in truth it does not rejecting H0 when H0 is in Fact True Investigators reject H0 and declare that a real effect exists when the chance of this decision being wrong is less than 5%. Two Possible Errors of Hypothesis Testing**The Type II Error occurs when we conclude that there is no**difference between treatments when in truth there is a difference fail to reject H0 when H0 is in fact False Two Possible Errors of Hypothesis Testing**In many circumstances a type I error is often regarded as**more serious than a type II error. Example: H0: innocent vs. H1: guilty Type I error = declaring an innocent man guilty Type II error = declaring a guilty man innocent Presumption of innocence Negative test result means "There is not enough evidence to convict“ rather than "innocence" Two Possible Errors of Hypothesis Testing**One will never know whether one has committed either error**unless data are available for the entire population. The only thing we are able to do is to assign α and β as the probabilities of making either type of error. It is important to keep in mind the difference between the truth and the decision that is being made as a result of the experiment. Review of errors in hypothesis testing**Hypothesis testing**• Null Hypothesis • No difference between Treatment and Control • Type I error, alpha, , p-value • The probability of declaring a difference between treatment and control groups even though one does not exist (ie treatment is not statistically different from control in this experiment) • As this is “society’s risk” it is conventionally set at 0.05 (5%)**Hypothesis testing**• Type II error, beta, • The probability of not declaring a difference between treatment and control groups even though one does exist (ie treatment is statistically different from control in this experiment) • 1 - is the power of the study • Often set at 0.8 (80% power) however many companies use 0.9 (90% power) • Underpowered studies have less probability of showing a difference if one exists**Steps in Hypothesis Testing**• Choose the null hypothesis (H0) that is to be tested • Choose an alternative hypothesis (HA) that is of interest • Select a test statistic, define the rejection region for decision making about when to reject H0 • Draw a random sample by conducting a clinical trial**Steps in Hypothesis Testing**• Calculate the test statistic and its corresponding p-value • Make conclusion according to the pre-determined rule specified in step 3**Hypothesis Testing - How to test a hypothesis**• Assume that we believe that we have a fair coin – equal chance of getting H or T when we flip the coin • Test the hypothesis by carrying out an experiment.**Hypothesis Testing - How to test a hypothesis**• Flip the coin 4 times, each time is H. What is the likelihood of getting 4 H if this is a fair coin?**Remember the Binomial Probability Function**Let X be the event of getting a H X ~ Binomial (n = 4, p=0.5) In this case, we want x=4 = 0.0625 = 6.25%