Inferential statistics

• Hypothesis is a statement or conjecture about any phenomena for which the truth has not been verified
• Hypothesis will need investigation to ascertain its truthfulness
• Examples of hypotheses:

  • Fear increases hypertension
  • Females are more prone to rheumatoid arthritis
  • Breast cancer is commoner in women who do not breastfeed
  • Sore throat is commoner in winter
• Procedure for test of hypothesis:

  • State Null Hypothesis
  • State alternative hypothesis
  • State level of statistical error
  • Choose appropriate test statistic
  • Apply data and evaluate test statistic
  • Take decision: reject or not reject null hypothesis
• Null Hypothesis:

  • Denoted by H_o
  • It is the hypothesis under test
  • It is the hypothesis to be nullified if data does not support it
  • Example: No difference in age of onset of rheumatoid arthritis between males and females
• Alternative Hypothesis:

  • Denoted by H_a
  • It is the hypothesis to consider if H_o is rejected
  • Statement in the affirmative language
  • Example: There is a difference in the age of onset of Rheumatoid Arthritis between males and females
• Level of Statistical Error:

  • Alpha or type 1 error (α): This is the error committed for rejecting true null hypothesis
  • Always very small (0.05 or 5%)
• Type of Errors in Decision Making:

  • Type 1 Error (α): Reject a true Null Hypothesis. This is rejecting a Null Hypothesis when it is indeed True
  • Type II Error (β): Failure to reject a false Null Hypothesis. This is not rejecting a Null Hypothesis when it is indeed false (β-ERROR)
• The P-Value:

  • Type 1 error is the level of significance – (α) (alpha)
  • Type II error is denoted by β (beta)
  • 1 - β = Power of the test
• Usual Objectives of Studies:

  • Compare characteristics of groups particularly average values
  • Investigation of association or relationship between variables
• Choice of Test Statistic:

  • Depend on study objectives (research questions)
  • Kind of data (quantitative or qualitative)
  • Sample size (small or large)
• Type of Test Statistics:

  • Parametric tests
  • Non-parametric tests
• Parametric Tests:

  • Assume distributional forms for the measurements and parameters in the population
  • Commonest assumption is normal distribution
  • Examples:
  • Z-test for comparing 2 proportions
  • T-test to compare mean values between only 2 groups
  • F-test or analysis of variance to compare mean values between several groups (more than 2 groups)
• Non-Parametric Tests:

  • Do not assume any particular functional form for a population distribution
  • Called distribution-free methods
  • Examples:
  • CHI-SQUARE TEST
  • MANN-WHITNEY-U TEST
  • WILCOXON SIGNED RANK SUM TEST
  • KRUSKAL-WALLIS TEST
  • MEDIAN TEST
• Statistics is a field of study concerned with:

  • The collection, organization, summarization, and analysis of data
  • The drawing of inferences about a body of data when only a part of the data is observed
• Statistics can be broadly classified into 2 categories:

  • Descriptive statistics
  • Inferential statistics
• Descriptive statistics include:

  • Measures of central tendency
  • Measures of dispersion
  • Diagrammatic (Graphic or pictorial) presentation of data
• Inferential statistics deals with extrapolation, drawing conclusions to the population based on observations from a sample
• Population:

  • The largest collection of entities for which we have an interest, which may be finite or infinite
• Sample:

  • A part of a population that is analyzed to draw conclusions about the population
• Parameter vs. Statistic:

  • When information is based on the entire population, it is a parameter
  • When information is based on a sample, it is a statistic or an estimate
• Study statistics are estimates of corresponding population parameters because the true values of the population are usually unknown
• The purpose of inferential statistics is to determine whether relationships observed in a sample are "real" or due to chance variation
• Hypothesis:

  • A claim or assumption about the population parameter, such as the population mean or proportion
• Null Hypothesis (H0):

  • The hypothesis under test that is nullified if data does not support it
  • It is a statement of no difference, association, effect, or equality
• Alternative Hypothesis (Ha):

  • The research question considered if the null hypothesis is rejected
  • It is a statement in affirmative language
• Errors in Hypothesis Tests:

  • Type I error: Rejecting the null hypothesis when it is true (α)
  • Type II error: Failing to reject the null hypothesis when it is false
• Type I error is the error committed when you reject the null hypothesis, but in reality, the null hypothesis is correct
• Type I error is also known as alpha (α) error
• Type I error is made if you find an association where there is none, or if you claim there is a difference when there isn't, or if you state a treatment has an effect when it does not
• Type II error is the error committed when you fail to reject the null hypothesis, but in reality, the null hypothesis is false
• Type II error is also known as beta (β) error
• Type II error is made if you find no association where there is one, or if you claim there is no difference when there is, or if you state a treatment has no effect when it does
• Level of statistical error, also known as the level of significance or alpha (α) error, is the maximum probability of committing a Type I error
• A small p-value (less than 0.05) leads to the rejection of the null hypothesis
• Power (1-β) of a test is its ability to reject a null hypothesis when it is false
• 1. Value is a measure of how much evidence we have against the null hypothesis
• Smaller p-value implies greater inconsistency
• Conventionally, we reject the null hypothesis if the p-value is less than 0.05
• Steps in testing of statistical hypothesis: