Parametric Methods in Statistics
Parametric statistical methods are those that make assumptions regarding the distribution of the population. These methods presume that the data have a known distribution (e.g., normal, binomial, Poisson) and rely on parameters (e.g., mean and variance) to define the data.
Key Assumptions
Parametric techniques make some assumptions regarding the population data:
- Distributional Assumption: Data should have a particular probability distribution (e.g., normal distribution in regression models).
- Independence: Observations should be independent of one another.
- Homogeneity of Variance: The variance of data should be equal across different groups or conditions.
- Large Sample Size: Most parametric techniques assume that the sample size is sufficiently large to estimate the population distribution closely.
Common Parametric Models
1. Normal Distribution Model: The normal distribution is the basis of most parametric techniques. It has a bell-shaped curve and is completely specified by two parameters: mean (μ) and variance (σ²). Most statistical tests, including t-tests and ANOVA, require normality.
2. Linear Regression: Linear regression describes the relationship between a dependent variable and one or more independent variables by assuming a linear relationship.
3. Logistic Regression: Logistic regression applies to binary classification problems. Rather than modeling the dependent variable directly, it models the probability that an observation is in a given category.
4. Poisson Regression: Applied to count data, Poisson regression assumes the dependent variable follows a Poisson distribution. It is widely applied in event modeling and time-series analysis.
5. Analysis of Variance (ANOVA): ANOVA performs tests for differences between means in more than two groups. It makes the assumption that the data are drawn from normally distributed populations with equal variances.
6. Bayesian Inference: Bayesian approaches apply prior distributions to revise estimates about a parameter on the basis of observed data.
Parameter Estimation Techniques
1. Maximum Likelihood Estimation (MLE): MLE determines the parameter values which maximize the likelihood function. For a normal distribution, the MLE for the mean is u and for variance
2. Method of Moments: This method estimates parameters by equating sample moments to population moments.
3. Least Squares Estimation: Applicable to regression models, least squares minimize the sum of squared residuals to estimate parameters.
Hypothesis Testing in Parametric Methods
1. t-Test
t-test is utilized to compare one or two group means and identify if the found differences are significant statistically. It works best with small sample size and normally distributed data.
Types of t-Tests
a. One-Sample t-Test: This test is used to compare the mean of a single sample to a known population mean.
- Null Hypothesis (H0): The sample mean is equal to the population mean.
- Alternative Hypothesis (H1): The sample mean is different from the population mean.
The test statistic is calculated as:
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
Where:
\bar{x} = Sample mean: The average of the sample data.- μ = Population mean: The hypothesized mean of the population.
- s = Sample standard deviation: The standard deviation calculated from the sample.
- n = Sample size: The number of data points in the sample.
b. Independent (Two-Sample) t-Test: Used when comparing the means of two independent groups.
- Null Hypothesis (H0): The means of both groups are equal.
- Alternative Hypothesis(H1): The means of both groups are different.
The test statistic is:
t = \frac{\bar{x_1} - \bar{x_2}}{\sqrt{s_1^2/n_1 + s_2^2/n_2}}
c. Paired t-Test: Used when comparing means of the same group before and after treatment.
t = \frac{\bar{d}}{s_d / \sqrt{n}}
Where:
\bar{d} = Mean of the differences between paired observations: The average of the differences between corresponding pairs of values (e.g., post-test score - pre-test score).- sd = Standard deviation of the differences between paired observations.
- n = Number of paired observations (sample size).
2. F-Test
An F-test is used to compare the variances of two or more groups. It is commonly used in Analysis of Variance (ANOVA) to determine if there is a significant difference between the means of multiple groups.
One-Way ANOVA: Used when comparing the means of three or more independent groups.
- Null Hypothesis (H0): All group means are equal.
- Alternative Hypothesis (H1): At least one group mean is different.
The F-statistic is given by:
F = \frac{\text{Between-group variance}}{\text{Within-group variance}}
where:
- Between-group variance: Measures how much the group means deviate from the overall mean.
- Within-group variance: Measures the variation within each group.
If the F-statistic is large, it indicates that at least one group is significantly different.
Two-Way ANOVA: Used when testing the effect of two independent factors on a dependent variable.
- Helps analyze interaction effects between two factors.
- Requires multiple groups for comparison.
3. Chi-Square Test
A chi-square test is used for categorical data to determine if two variables are independent. It compares observed frequencies with expected frequencies under the assumption of independence.
Chi-Square Test for Independence
Used to test the association between two categorical variables.
- Null Hypothesis (H0)-The two variables are independent.
- Alternative Hypothesis (H1)- The two variables are dependent.
The test statistic is:
\chi^2 = \sum \frac{(O - E)^2}{E}
Where:
- O = Observed frequency: The actual frequency observed in the data.
- E = Expected frequency: The frequency that is expected under the null hypothesis or under some theoretical model.
4. Likelihood Ratio Test
The likelihood ratio test is used to compare the goodness-of-fit between two statistical models: one is a simpler (null) model, and the other is a more complex (alternative) model.
How It Works
- Calculate the likelihood function for both models.
- Compute the likelihood ratio statistic:
- The test statistic follows a chi-square distribution:
- Compare the statistic with a chi-square critical value to determine significance.
Advantages of Parametric Methods
- More powerful than non-parametric tests when assumptions are met.
- Offer exact estimates of parameters.
- Good theory and interpretation established.
- Efficient for large data sets.
Limitations of Parametric Method
- Need to make strong assumptions regarding data distribution.
- Sensitive to model misspecification and outliers.
- May not be appropriate for small datasets with unobserved distributions.
Applications of Parametric Methods
1. Economics: Applied in regression models to study market trends and consumer behavior.
2. Machine Learning: Most supervised learning algorithms, including logistic regression and Bayesian inference, are based on parametric methods.
3. Medicine: Applied in clinical trials and epidemiology to study treatment effects.
4. Engineering: Applied in quality control and reliability analysis.