Frequentist Statistics Overview
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Questions and Answers

What is the value of the parameter lambda in the Poisson distribution?

  • Neither mean nor variance
  • Both mean and variance (correct)
  • Mean only
  • Variance only

    • The Poisson distribution is classified as a continuous distribution.

    False

    What kind of probability does the Binomial distribution calculate?

    The probability of seeing an event happen a certain number of times.

    The Binomial distribution is commonly used in bioinformatics and can provide the probability of detecting ___ sequence variants in a population.

    <p>sequence</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Lambda = Parameter in the Poisson distribution n = Number of trials or observations k = Number of successes p = Probability of success on each trial</p> Signup and view all the answers

    In the expression for the Binomial distribution p(x = k), which element represents the number of successes?

    <p>k</p> Signup and view all the answers

    The Binomial distribution can only be applied to events that occur two times.

    <p>False</p> Signup and view all the answers

    What mathematical notation is commonly associated with the Binomial distribution?

    <p>n choose k notation (nCk)</p> Signup and view all the answers

    What does the likelihood function provide?

    <p>The probability that a certain sample was drawn from a distribution</p> Signup and view all the answers

    The maximum likelihood estimate can provide the true value of the distribution parameters.

    <p>False</p> Signup and view all the answers

    What happens to the degrees of freedom when a parameter is estimated from a sample?

    <p>One degree of freedom is used up.</p> Signup and view all the answers

    The total degrees of freedom of a random sample is equal to the number of data points in it minus the number of _______ estimated parameters.

    <p>parameters</p> Signup and view all the answers

    Which statement correctly describes the relationship between data points and estimated parameters?

    <p>At least one data point is needed for each parameter estimated.</p> Signup and view all the answers

    Estimating parameters from a sample always increases the available data points.

    <p>False</p> Signup and view all the answers

    What is the relationship between the population estimate and the maximum likelihood estimate?

    <p>The population estimate is the true parameter value, while the maximum likelihood estimate is an estimation based on sample data.</p> Signup and view all the answers

    Match the following concepts with their definitions:

    <p>Likelihood Function = Gives probability that a sample came from a specific distribution Maximum Likelihood Estimate = Estimate derived from a sample to maximize likelihoods Degrees of Freedom = Number of independent pieces of information used in estimation Population Estimate = True parameter value of the entire population</p> Signup and view all the answers

    What is the main topic discussed in the content?

    <p>Philosophy of statistics</p> Signup and view all the answers

    Michael's argument that there can be no frequentist probability for a single event, like a volcano erupting tomorrow, is correct.

    <p>False</p> Signup and view all the answers

    What is the significance of the probability values mentioned for the two distributions?

    <p>They indicate the likelihood of the measurements belonging to each distribution.</p> Signup and view all the answers

    The formula for the function f(x) is given by f(x) = c e^{____(x - µ)^2 / (2s^2)}.

    <p>-(x - µ)^2</p> Signup and view all the answers

    If the average blood pressures from two professors are being analyzed, what is likely being tested?

    <p>The difference between the two averages</p> Signup and view all the answers

    According to the content, there are definitive right or wrong answers to the exercises.

    <p>False</p> Signup and view all the answers

    In the scenario presented, what percentage chance is attributed to the possibility of Mount St Helens erupting tomorrow?

    <p>10%</p> Signup and view all the answers

    What does SEdi f f measure?

    <p>The variability in the difference between two sample means</p> Signup and view all the answers

    An increase in variability within samples leads to a lower SEdi f f.

    <p>False</p> Signup and view all the answers

    What is the threshold for determining statistical significance in the context given?

    <p>0.05</p> Signup and view all the answers

    The variance of samples x and y is denoted by ______ and ______ respectively.

    <p>s2x, s2y</p> Signup and view all the answers

    Match the following components of the t-test with their descriptions:

    <p>s2x = Variance of sample x s2y = Variance of sample y nx = Number of observations in sample x ny = Number of observations in sample y</p> Signup and view all the answers

    What happens when you divide the variances by the number of observations?

    <p>It provides a fair estimate of the variance per sample.</p> Signup and view all the answers

    A t-value can be deemed sufficiently large if it is more extreme than 1% of the t-distribution.

    <p>False</p> Signup and view all the answers

    In frequentist statistics, what does probability represent?

    <p>Frequency of occurrence</p> Signup and view all the answers

    What function is used to estimate the likelihood of data drawn from a normal distribution?

    <p>dnorm</p> Signup and view all the answers

    Tossing a coin ten times and counting the number of heads is an example of a binomial distribution.

    <p>True</p> Signup and view all the answers

    What is the purpose of calculating a 95% confidence interval in statistical analysis?

    <p>To estimate the range within which the true population parameter is likely to fall with 95% certainty.</p> Signup and view all the answers

    The joint probability of independent events can be calculated by using the _______ of their individual probabilities.

    <p>product</p> Signup and view all the answers

    Match each statistical concept with its definition:

    <p>Likelihood = Probability of obtaining the observed data given a statistical model Confidence Interval = Range of values within which a population parameter lies with a specified probability Binomial Distribution = Distribution of the number of successes in a fixed number of trials Normal Distribution = Continuous probability distribution characterized by its bell-shaped curve</p> Signup and view all the answers

    In the function rbinom(n=10, size=1, prob=0.5), what does 'size=1' represent?

    <p>Number of coins tossed at each trial</p> Signup and view all the answers

    The average systolic blood pressures of the professors are necessary to calculate maximum likelihood estimates.

    <p>True</p> Signup and view all the answers

    What value is given as a mean for estimating likelihood based on the provided data?

    <p>120</p> Signup and view all the answers

    What is the primary reason for calculating the density of the t-distribution?

    <p>To rank observed t-values</p> Signup and view all the answers

    A one-tailed test is preferred when you are only interested in one direction of data.

    <p>True</p> Signup and view all the answers

    What quantiles are checked in a two-tailed test at a significance level of 0.05?

    <p>2.5th and 97.5th quantiles</p> Signup and view all the answers

    When conducting a one-tailed test, the risk of incorrectly detecting a significant difference is _____.

    <p>increased</p> Signup and view all the answers

    What is the likelihood of detecting a true difference at a significance level of 0.05?

    <p>80%</p> Signup and view all the answers

    What do 95% confidence intervals represent in relation to a significance level of 0.05?

    <p>Regions where differences by chance would be surprising.</p> Signup and view all the answers

    What is the danger of performing a one-tailed test just to achieve significance?

    <p>It may lead to false positives.</p> Signup and view all the answers

    Match the following terms with their correct descriptions:

    <p>Two-tailed test = Tests both lower and upper tails of the distribution One-tailed test = Tests only one direction for significance Significance level 0.05 = Represents a 5% chance of making a Type I error Confidence interval = Range where differences are not due to chance</p> Signup and view all the answers

    Study Notes

    Part I: Frequentist Statistics

    • Frequentist statistics is a branch of statistics
    • Probability is defined as a fraction of times an outcome is expected in the long run.
    • This is different from Bayesian philosophy, which views probability as a belief.

    Session 1: Philosophy of Statistics

    • Statistics is a field broader than bioinformatics.
    • Probability definitions vary among statisticians.
    • Frequentist probability is based on observed outcomes, whereas Bayesian probability is based on beliefs.
    • Statistical distributions describe relative frequencies of sample values.
    • An example is the distribution of heights of college students.
    • Some distributions are continuous, like height, while others are discrete, like integers.
    • The normal distribution is a frequently used continuous distribution.

    Session 1: Estimating parameters

    • Maximum Likelihood Estimation (MLE) finds the parameters of a distribution that maximizes the probability of observing the given data.
    • The Central Limit Theorem describes the tendency for sample means to approximate a normal distribution under certain conditions.
    • The normal distribution is commonly used in statistics due to the Central Limit Theorem.
    • Estimating parameters involves considering the unknown parameters of a distribution from which a sample was drawn.
    • It’s estimated through the sample to generate an estimate of the parameters of a true distribution.

    1.2 Statistical Distributions

    • Statistical distributions describe the relative frequencies with which different values of a variable are drawn.
    • Continuos distributions describe values of ranges within them, example: distributions of continuous data (height of individuals)
    • Discrete distributions describe individual values of data, e.g., counts of events (e.g., number of heads when flipping a coin 100 times)
    • The normal distribution, described by mean and standard deviation, is a frequently encountered distribution
    • The normal (or gaussian) distribution is a particular case of all the statistical distributions

    1.3 Estimating Parameters: Maximum Likelihood Estimation

    • A likelihood function is a way to measure the probability of a set of data given a set of parameters.
    • The maximum likelihood estimate (MLE) is a statistical parameter estimate that maximizes a likelihood function given observed data points.
    • Degrees of freedom in an estimate refers to the number of observations remaining after the estimate is calculated.
    • Maximum likelihood is an estimate of the parameters that maximize the product of the probability of the data points

    1.4 The Zoo of Statistical Distributions

    • The normal distribution and many other statistical distributions are used frequently, and many are derived from others
    • Chi-Squared distribution relates to variation in data

    Session 2: Test statistics

    • Test statistics are used to determine whether an observation is "surprising" relative to expected values given certain assumptions
    • The t-test is used to compare the means of two groups and is sensitive to how variable the given data set is
    • If two sets of data are drawn from the same distribution then the expected difference will be 0
    • The t-statistic is the difference in the sample means divided by the standard error
    • The standard error is a measure of how variable the given data set is
    • The p-value shows how often a result similar to or more extreme than the measured value of the t-statistic would have been observed if there was no actual difference between the two distributions.

    Session 2: Regression and ANOVA

    • Linear regression models a relationship between a response and explanatory variable
    • ANOVA extends linear regression to use multiple explanatory variables, which are categories instead of just numbers
    • Both models fit data by minimizing squared error (prediction error for linear case, or 'error' in groups by ANOVA)

    Session 3: Multiple Models

    • Multiple regression (and ANOVA) are extensions of linear regression for cases where multiple explanatory variables are used.
    • Multiple response variables require multivariate regression methods.
    • Model criticism examines how well a model is fitting by checking its assumptions about the data
    • The r squared value describes the amount of variation in the data that the model can explain

    Session 4: Hierarchical (Bayesian) Models

    • Hierarchical Models are a more detailed way of analysing models
    • Random effects are a way of analyzing variation in data that comes from different sources, e.g., different individuals or trials.
    • Bayesian methods take account of uncertainty or beliefs about parameters by providing a range of possible parameter values with associated probabilities
    • Using a Bayesian approach or hierarchical modelling is suitable for analysing data with uncertain effects and multiple sources of random variation

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    Description

    Explore the foundational concepts of frequentist statistics, focusing on its philosophy and the interpretation of probability. Learn about statistical distributions, parameter estimation through Maximum Likelihood Estimation, and how these concepts differ from Bayesian statistics.

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