Parametric Inference with R

Parametric inference in R involves the process of drawing statistical conclusions regarding a population using a parametric statistical framework. These parametric models make the assumption that the data adheres to a specific probability distribution, such as the normal, binomial, or Poisson distributions, and they incorporate parameters to characterize these distributions.

It is a technique that involves making assumptions, about the probability distribution underlying your data. Based on these assumptions you can then draw conclusions. Make inferences about population parameters. In the R programming language parametric inference is frequently employed for tasks such, as hypothesis testing and estimating parameters.

General steps for executing parametric inference in R

• Data Collection: Begin by collecting and preparing your data. Data can be imported into R using various functions like read.csv() and read.table().
• Exploratory Data Analysis (EDA): Before proceeding to fit a parametric model, it is imperative to gain a comprehensive understanding of your data. Employ R functions like summary(), hist(), boxplot(), and scatterplots to visualize and summarize your dataset.
• Selection of a Parametric Model: Based on the insights gleaned during EDA, opt for a suitable parametric model that best characterizes the data’s distribution. For example, if your data exhibits characteristics resembling a normal distribution, you may opt for the normal distribution model.
• Parameter Estimation: Proceed to estimate the parameters intrinsic to the chosen parametric model from your dataset. R offers a variety of functions such as mean(), var(), and glm() (for more complex models) to facilitate parameter estimation.
• Hypothesis Testing: Conduct hypothesis tests to derive inferences regarding population parameters. Common hypothesis tests encompass t-tests, chi-squared tests, ANOVA, and others. R provides dedicated functions like t.test(), chisq.test(), and anova() for executing these tests.
• Confidence Intervals: Calculate confidence intervals to ascertain the probable ranges for population parameters. You can employ functions like confint() or craft custom code to accomplish this task.
• Model Assessment: Thoroughly evaluate the appropriateness of your parametric model’s fit to the data. This assessment involves utilizing diagnostic plots, conducting residual analysis, and applying goodness-of-fit tests.
• Drawing Inferences: Based on the outcomes of your analysis, formulate inferences pertaining to the underlying population. These inferences could include statements such as “there is substantial evidence to suggest that the population mean likely falls within a specific range” or “there exists statistical significance indicating a noteworthy difference between groups

Parametric Inference for two-sample T-test

R

# Example: Performing a two-sample t-test
# Assuming you have two sets of data in vectors x and y

# Create sample data (replace this with your actual data)
x <- c(25, 30, 35, 40, 45)
y <- c(22, 27, 33, 38, 41)

# Conduct a two-sample t-test
result <- t.test(x, y)

# Print the results
print(result)

# Extract specific values like the p-value and confidence intervals
p_value <- result$p.value
conf_int <- result$conf.int
cat('p-value :',p_value,'\n')
cat('Confidence Interval :',conf_int)

Output:

    Welch Two Sample t-test
data:  x and y
t = 0.56408, df = 7.9983, p-value = 0.5882
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -8.647129 14.247129
sample estimates:
mean of x mean of y 
     35.0      32.2 
p-value : 0.5881658 
Confidence Interval : -8.647129 14.24713

• The two-sample t-test findings, including the t-statistic, degrees of freedom, p-value, and confidence intervals, are computed using the t.test function. The assumption that the means of the two samples are equivalent is put to the test.
• The result object prints a summary of the t-test when it is printed. This includes details about the alternative hypothesis, which is often that the means are not equal, the p-value, the confidence interval for the difference in means, and the t-statistic and its related degrees of freedom.
• The p-value obtained from the t-test is stored in the p_value variable. Under the null hypothesis, it estimates the likelihood of discovering the observed difference in means (or a more extreme difference). Stronger evidence that the null hypothesis is incorrect is shown by lower p-values.
• The confidence interval for the mean difference is kept in the conf_int variable. It provides a range, with a defined level of confidence (often 95% by default), within which the genuine difference in means is expected to fall.

Here, let’s dive deeper into the concept of parametric inference in R with examples.

Parametric Inference for Normal Distribution:

Lets say you have a set of exam scores. You want to check if they follow a distribution. To do this you can utilize the Shapiro Wilk test, which’s a tool, in R. Here’s how you can perform parametric inference in R to assess normality:

R

# Sample data
exam_scores <- c(85, 92, 78, 88, 95, 90, 87, 89, 82, 91)

# Shapiro-Wilk test for normality
shapiro.test(exam_scores)

Output:

Shapiro-Wilk normality test
data:  exam_scores
W = 0.96917, p-value = 0.883

By using the function you can test the hypothesis that the data conforms to a distribution. The results will provide a test statistic and a p value. If the p value is lower than your chosen significance level (, for example 0.05) it would imply rejecting the hypothesis and suggesting that the data does not exhibit a distribution.

Parametric Inference for Mean Comparison (T-Test):

Suppose you possess data derived from two distinct groups, namely Group A and Group B, and you desire to ascertain whether a noteworthy disparity exists in the averages of these two groups. To adequately address this concern, employing a t-test is fundamental, predicated on the prerequisite that the data within each group adheres to a Gaussian distribution. Thus, allow me to present to you the procedural steps required to execute this parametric inference within the R programming language.

R

# Sample data for Group A and Group B
group_a <- c(28, 30, 32, 35, 27)
group_b <- c(24, 26, 29, 30, 28)

# Perform a two-sample t-test
t_test_result <- t.test(group_a, group_b)

# Print the results
print(t_test_result)

Output:

Welch Two Sample t-test
data:  group_a and group_b
t = 1.6718, df = 7.4203, p-value = 0.136
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -1.194927  7.194927
sample estimates:
mean of x mean of y 
     30.4      27.4 

By employing the t.test function, one can execute a two-sample t-test predicated on the assumption of data normality. This function yields an output inclusive of the test statistic, degrees of freedom, and the p-value. If the calculated p-value falls below the prespecified significance level, it is possible to deduce that there exists a significant dissimilarity in the means exhibited by the two respective groups.

Parametric Inference for Linear Regression

Parametric inference is commonly used in analyzing regression. Lets say you have a dataset, with two variables. You want to create a regression model that predicts one variable based on the other. Here’s an example:

R

# Sample data
x <- c(1, 2, 3, 4, 5)
y <- c(2, 3, 4, 5, 6)

# Fit a linear regression model
model <- lm(y ~ x)

# Summary of the regression model
summary(model)

Output:

Call:
lm(formula = y ~ x)
Residuals:
         1          2          3          4          5 
 3.395e-16 -3.543e-16 -1.716e-16  4.793e-17  1.384e-16 
Coefficients:
            Estimate Std. Error   t value Pr(>|t|)    
(Intercept) 1.00e+00   3.27e-16 3.058e+15   <2e-16 ***
x           1.00e+00   9.86e-17 1.014e+16   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 3.118e-16 on 3 degrees of freedom
Multiple R-squared:      1,    Adjusted R-squared:      1 
F-statistic: 1.029e+32 on 1 and 3 DF,  p-value: < 2.2e-16

To fit the linear regression model the lm function is utilized. This parametric inference technique assumes a linear relationship between x and y. Estimates the coefficients (intercept and slope) of the equation. The summary function provides information about the model, including estimates, standard errors, t values and p values.

Conclusion

In these examples we have explored how parametric inference can be applied in R for testing assumptions (such as normality) comparing means and conducting linear regression analysis. However it is important to keep in mind that the validity of tests relies on assumptions about the data distribution. If these assumptions are not met alternative methods, like parametric approaches or data transformation may be more suitable.