Estimation and Hypothesis Testing

• Estimation is about inferring properties of a dataset's underlying distribution, using inductive rather than deductive reasoning.
• Statistical inferences can be of two types: estimation and hypothesis testing.
• Estimation involves estimation of specific values, known as point estimates; interval estimation specifies a range for the likely values.
• Hypothesis testing checks whether a population parameter's value equals a specific value.

Understanding Population and Sample Relationships

• A random sample requires each member to be independently chosen with a known probability of selection.
• Simple random samples ensure each member has the same selection probability.
• The reference population (target or study population) is the group of interest.
• Cluster sampling is an alternative to random sampling.

Exploring Random-Number Tables

• A random number or digit is a random variable X taking on values 1, 2,... 9 with equal probability, such as Pr(X = 0) = Pr(X = 1) = ... = Pr(X = 9) = 1/10.
• Computer-generated random numbers have two properties: each digit is equally likely to occur, and each digit's value is independent of others.
• Computer programs can generate large sequences of random digits, referred to as pseudo-random numbers because they simulate randomness.

Random Selection and Assignment

• In an example, each of 1000 participants is assigned a number from 000 to 999, with groups of three digits selected from a random number table.
• Random assignment differs from random selection because the number is fixed in advance.
• In a finite sample, random assignment is better, because it ensures equal patient assignment to each group.

Utilizing Randomized Clinical Trials (RCTs)

• Randomized clinical trials (RCTs) are considered the optimal study design for clinical research.
• RCTs compare different treatments by randomly assigning particular treatments to patients.
• The process of assigning treatments to patients is called randomization.
• Randomization ensures that different treatment modalities will be similar, especially with large sample sizes; if sample sizes are small, patient characteristics in treatment groups could be incomparable.
• RCTs usually include a table of treatment group characteristics to verify that the randomization process works well.

Design Features of Randomized Clinical Trials

• Methods for randomization include random selection and random assignment, the latter sometimes called block randomization in clinical trials.
• With a block size of 2n, n patients are randomly assigned to treatment A, and the remaining n to treatment B.
• For "k" treatment groups, the block size is kn, where "n" patients are randomly assigned to each of the "k" treatments.

Stratification Techniques

• It is a technique used in the randomization process.
• In clinical studies, patients are subdivided into subgroups, or strata, based on relevant characteristics.
• Separate randomization lists are maintained for each stratum to ensure comparable patient populations within each stratum.
• Either standard random selection or block randomization can be used in each stratum.
• Typical characteristics for defining strata are age, sex, and overall clinical condition.

Blinding in Trials

• Blinding is an important aspect of modern clinical research.
• In a double-blind clinical trial, neither the physician nor the patient knows the treatment being given.
• In single-blind trials, patients are unaware of their treatment assignment, while physicians are.
• In unblinded trials, both the physician and the patient are aware of the treatment assignment.
• The gold standard is the randomized double-blind study, where patients are randomly assigned treatment and neither the patient nor physician knows the assignment.
• This is to prevent biased reporting of outcomes.
• Since in some cases treatment side effects may strongly suggest the actual treatment, it is not always possible.

Estimation of the Mean of a Distribution

• The natural estimator for the population mean (µ) is the sample mean (X), calculated as the sum of "n" data points "Xi" divided by "n".
• x is a single realization of a random variable X over all possible samples of size n.
• The sampling distribution of x is the distribution of possible x values across all samples of size n.
• The frequency distribution illustrates the sample mean from randomly selected samples.
• The expected value of X over its sampling distribution is equal to μ.

Unbiased Estimators

• If X₁, ..., Xₙ is a random sample from a population with mean µ, then the sample mean X, with E(X) = µ, regardless of the underlying distribution.
• For symmetric distributions, multiple unbiased estimators of "mu" exist, like the sample median.
• The sample mean is chosen commonly because with a normal distribution, it is the unbiased estimator with the smallest variance.

Standard Error Estimation

• If X₁, ..., Xₙ is a random sample from a population with mean μ and variance σ², the set of sample means in repeated samples of size n has variance σ²/n.
• The standard deviation of sample means is σ/√n, defined as the standard error of the mean or the standard error.
• An estimator of population variance σ² is the sample variance s².
• The standard error of the mean (SEM or SE) is σ/√n, which is estimated as s/√n.
• It represents the estimated standard deviation of sample means from repeated samples with underlying variance σ².
• The standard error quantifies the variability of sample means from repeated random samples.
• It is directly proportional to 1/√n and to population standard deviation σ of individual observations.
• Precision in estimating μ is affected by variance σ² and sample size.

Central Limit Theorem

• If X₁, ..., Xₙ is a random sample from a population with mean µ and variance σ², for a large enough "n", distribution of X approximates a normal distribution N(µ, σ²/n).
• This normality justifies statistical inferences even if individual observations are non-normally distributed.
• Data can be transformed by using log scales to reduce skewness and allow the theorem to apply to smaller sample sizes.

Interval Estimation

• Interval estimation specifies a range within which parameter values will likely fall.
• Re-expressing X in standardized form by Z = (X - μ) / (σ / √n), then follows a standard normal distribution.
• Approximately 95% of "Z" values from repeated samples fall between -1.96 and +1.96.
• In practice, σ is rarely known.

Confidence Intervals

• It is an approximate formula is used to get a 100% × (1 - α) confidence interval (CI) for the mean µ of a normally distributed population, when the variance is unknown.
• It is given by the range (x - z1-α/2 * s/√n, x + z1-α/2 * s/√n).
• This interval is valid only if n > 200.
• It can also be used for n ≤ 200 if "sigma" is known by replacing "s" with "sigma".

Factors Affecting Confidence Interval Length

• Cls determine how precise the estimation is.
• The factors affecting it include:
• Sample Size: increased with larger sample sizes
• Deviation: increase with standard deviation
• Confidence: increases with desired confidence

Estimation of the Variance of a Distribution

• The sample variance S² is an unbiased estimator of σ². This applies to all random samples from this population, which is denoted as E(S²) = σ².
• The higher the variance, the greater the interobserver variability.