What is this? Why is this important
We are trying to predict or demonstrate a relationship between two observed quantities (dependent & independent).
As one quantity changes (A) in two dimensions how can we model this relationship to (B) as a line ? Resulting in (A,B) coortinates in a Cartesian plane.
What are these coordiantes (Observed values)
Typically it is what you are interested in. For instance how does the size of an organization impact money spent in a SaaS product.
But we are interested in predicting how each member added impacts dollars spent. So we use a line to assess incremental impact. A slope. A rise over run.
How is the line drawn in Two dimensions The least squares regression line is a line that best fits the data points in a way that minimizes the sum of the squared vertical distances between the line and the points
the smaller the standard error the better you model is at predicting incremental dollar gain per new org member. The larger the standard error perhaps the worse.
OLS The goal of OLS estimation is to find the values of the coefficients that minimize the sum of squared residuals (SSres) between the predicted values and the actual values in the dataset.
What is a coefficient A coefficient represents the change in the response variable that is associated with a one-unit change in the predictor variable, while holding all other predictor variables constant. In a linear regression model, the coefficient represents the slope of the regression line and is often used to make predictions about the value of the response variable based on the value of the predictor variable.
Y = β0 + β1*X + ε
SSE = ∑(Yi - Ŷi)^2 // how much is prediction off
To estimate the coefficients, we can use the following formulas:
β1 = ∑(Xi - Xbar)(Yi - Ybar) / ∑(Xi - Xbar)^2 // slopish
β0 = Ybar - β1*Xbar
//where Xbar and Ybar are the sample means of X and Y, respectively.
The slope of a straight line is constant, so the derivative of a straight line with respect to its slope will always be a constant value. In the case of linear regression, the objective is to find the values of the regression coefficients (i.e., the intercept and slope of the line) that minimize the sum of squared errors.
Again SSE = ∑(Yi - Ŷi)^2
where Yi is the observed value of the response variable for the i-th observation, and Ŷi is the predicted value of the response variable based on the linear regression model, which is given by:
Ŷi = β0 + β1 * Xi
To find the values of β0 and β1 that minimize SSE, we can take partial derivatives of SSE with respect to β0 and β1, respectively, and set them equal to zero.
The partial derivative of SSE with respect to β0 can be written as:
∂SSE / ∂β0 = -2 ∑(Yi - Ŷi)
To find the value of β0 that minimizes SSE, we set this partial derivative equal to zero and solve for β0:
∂SSE / ∂β0 = -2 ∑(Yi - Ŷi) = 0
Solving for β0, we get:
β0 = Ybar - β1 * Xbar
where Ybar and Xbar are the sample means of Y and X, respectively.
Similarly, the partial derivative of SSE with respect to β1 can be written as:
∂SSE / ∂β1 = -2 ∑(Yi - Ŷi) * Xi
To find the value of β1 that minimizes SSE, we set this partial derivative equal to zero and solve for β1:
∂SSE / ∂β1 = -2 ∑(Yi - Ŷi) * Xi = 0
Epsilon (ε) typically refers to the residual errors in a regression model. Residual errors are the differences between the actual observed values of the response variable and the predicted values of the response variable from the regression model. In other words, epsilon is a vector of individual errors, one for each observation in the dataset. The residual errors are denoted by εi, where i represents the i-th observation.
On the other hand, SSE (sum of squared errors) is a measure of the overall goodness of fit of the regression model. It is the sum of the squared residual errors across all observations in the dataset, i.e., SSE = Σ(εi)², where Σ represents the sum over all observations.
In summary, epsilon represents the residual errors for each observation in the dataset, while SSE is a measure of the overall goodness of fit of the regression model based on the sum of the squared residual errors. In summary, correlated errors can occur when there are systematic patterns in the residuals that are related to some other variable in the dataset.
When the assumption of independent errors holds, the ordinary least squares (OLS) method for minimizing SSE is valid and efficient. OLS is a method that estimates the values of the regression coefficients by minimizing the sum of squared errors, which is a measure of the difference between the observed values and the predicted values of the response variable.
If your residuals are still correlated is your predictor variable(s) really doing a good job predicting? Is your model valid?
There are several situations where the residuals in a regression model might not be independent. Here's an example:
Suppose you are building a model to predict housing prices based on various features of a house, such as square footage, number of bedrooms, and location. You collect data from a real estate website and build a linear regression model to predict housing prices.
However, you notice that there is some correlation between the residuals of the model, which violates the assumption of independence. This could occur if the data from the real estate website is clustered by neighborhood, and there are unobserved neighborhood-level factors that affect housing prices, such as school quality or crime rates. If the model does not include these neighborhood-level factors, the residuals might be correlated within each neighborhood, leading to violation of the assumption of independence.
How well does your model predict across the entire range of possible observed values?
Heteroscedasticity can occur in various real-world scenarios where the variance of the error terms changes across the range of the independent variables. Here's an example:
Suppose you are analyzing the relationship between a person's age (independent variable) and their income (dependent variable). You collect data from a sample of individuals ranging in age from 18 to 65. You run a linear regression model to estimate the effect of age on income.
If the variance of the error term is not constant across all values of age, you may observe heteroscedasticity in the data. For example, you may find that the variance of the errors increases as age increases, indicating that the residuals are more spread out for older individuals. This could occur if older individuals have more diverse sources of income than younger individuals, leading to a wider range of income levels and greater variance in the errors.
The assumption of normality of the residuals.
When performing regression analysis, the estimated coefficients (also known as regression coefficients or beta coefficients) represent the estimated relationships between the independent variables and the dependent variable
To assess the uncertainty associated with these estimated coefficients, confidence intervals are constructed. A confidence interval provides a range of plausible values for the coefficient, along with an associated level of confidence. For example, a 95% confidence interval for a coefficient means that we are 95% confident that the true population coefficient lies within that interval.
In ordinary least squares (OLS) regression, the standard errors of the estimated coefficients are calculated based on the assumption that the errors (or residuals) follow a normal distribution with constant variance. These standard errors are then used to construct confidence intervals around the estimated coefficients.
The estimated coefficients are obtained from a single sample of data, and they are used to make inferences about the population parameters. However, because we only have one sample, there is uncertainty associated with the estimated coefficients. The sampling distribution helps us understand this uncertainty by considering the range of estimated coefficients that we would obtain if we were to repeat the sampling process multiple times.
The shape of the sampling distribution depends on various factors, including the properties of the data and the underlying assumptions of the regression model. Under the assumption of normality, the sampling distribution of the estimated coefficients is often assumed to be approximately normal, especially for large sample sizes. In such cases, we can use the properties of the normal distribution to construct confidence intervals and perform hypothesis tests.
But I thought we were using partial derivitives to compute coefficients that minimize something? How does this fit in with standard errors?
The partial derivatives are used to find the values of the regression coefficients that minimize the sum of squared residuals. Once the estimates of the coefficients are obtained, the standard errors are calculated to assess the uncertainty associated with these estimates.
The standard errors quantify the variability of the estimated coefficients and reflect the spread of the coefficient estimates around their true population values. They provide a measure of how much the estimated coefficients are likely to vary if the regression analysis were repeated on different samples from the same population.
The standard errors can be calculated using various methods, including the matrix algebra approach or the formula-based approach. The most common method is based on the assumption that the errors (residuals) are independently and normally distributed with constant variance. Under this assumption, the standard errors are estimated using the residuals and the properties of the observed data.
By estimating the standard errors, we obtain a measure of uncertainty associated with the estimated coefficients. These standard errors are then used in various statistical procedures, such as hypothesis testing and constructing confidence intervals, to make statistical inferences about the population parameters and assess the significance of the estimated relationships.
Constant variance and independence of residuals are related but distinct concepts in regression analysis.
Constant variance, also known as homoscedasticity, means that the variability of the residuals is consistent across all levels of the independent variables. In other words, the spread or dispersion of the residuals should be constant throughout the range of the predictors. This assumption is important because it ensures that the errors have a consistent level of variability and that the model's predictions are equally reliable across the entire range of the independent variables.
On the other hand, independence of residuals refers to the lack of correlation or dependence between the residuals. It assumes that the residuals for different observations are not systematically related to each other. In other words, the value of the residual for one observation does not provide any information about the value of the residual for another observation. Independence is crucial because it allows us to make valid statistical inferences, conduct hypothesis tests, and construct reliable confidence intervals.
precision and reliability of the estimated coefficients.
In the context of regression analysis, efficiency refers to the statistical efficiency of an estimator, specifically the regression coefficient estimates. An efficient estimator is one that achieves the smallest possible variance among all unbiased estimators.
In ordinary least squares (OLS) regression, the estimated coefficients are the least squares estimates that minimize the sum of squared residuals. *These estimates are unbiased, meaning that on average, they are equal to the true population values. However, among all unbiased estimators, the OLS estimates are also the most efficient when certain assumptions are met.
When the assumptions of OLS regression, such as linearity, independence, constant variance, and normality of errors, are met, the OLS estimators are not only unbiased but also the most efficient. In this case, they achieve the smallest possible variances, making them highly desirable for inference and hypothesis testing.
Normality helps establih a probabalistic range of possible coefficient values?
- Graphical
it's worth considering graphical methods, such as histograms, Q-Q plots, and kernel density plots, to visually assess the normality of residuals.
In hypothesis testing, we start with a null hypothesis, which is a statement we want to test. The test statistic is calculated based on the observed sample data and is designed to summarize the evidence against the null hypothesis.
A test statistic quantifies the evidence against the null hypothesis by summarizing the relationship between the observed data and the hypothesis being tested. It provides a standardized measure that allows for statistical decision-making in hypothesis testing.
The test statistic is chosen in such a way that it has a known distribution under the null hypothesis. This distribution is typically derived based on theoretical considerations or statistical assumptions. The choice of the test statistic depends on the nature of the hypothesis being tested and the specific statistical test being performed.
In hypothesis testing and confidence interval estimation, a t statistic is typically computed as the ratio of the difference between an estimate and a hypothesized value (usually the population mean or coefficient) to the standard error of the estimate.
The formula for a t statistic is:
t = (estimate - hypothesized value) / standard error
For example, in a t-test comparing the means of two independent groups, the t statistic is calculated as the difference between the sample means divided by the standard error of the difference. In regression analysis, t statistics are computed for each coefficient to test their significance by comparing the coefficient estimate to zero.
So in regression, the coefficients are the test statistics
No, in regression, the coefficients themselves are not considered test statistics. The coefficients in regression analysis represent the estimated parameters that describe the relationship between the independent variables and the dependent variable.
In hypothesis testing in regression analysis, the test statistics are typically derived from the coefficients to assess their significance. These test statistics are used to determine whether the estimated coefficients are statistically different from zero.
The most commonly used test statistic in regression analysis is the t-statistic. The t-statistic is calculated by dividing the estimated coefficient by its standard error. It measures the number of standard errors the estimated coefficient is away from zero.(no relationship)
The t-statistic follows a t-distribution, which has a specific shape determined by the degrees of freedom. The degrees of freedom depend on the sample size and the number of independent variables in the regression model.
To determine the significance of a coefficient, you compare the absolute value of the t-statistic to critical values from the t-distribution or calculate the associated p-value. If the absolute value of the t-statistic exceeds the critical value or if the p-value is smaller than the predetermined significance level (e.g., 0.05), the coefficient is considered statistically significant, indicating that it is unlikely to be zero in the population.
No, coefficients in regression analysis are not effect sizes. Effect sizes and coefficients serve different purposes and convey different information.
In regression analysis, coefficients represent the estimated parameters that quantify the relationship between the independent variables and the dependent variable. They indicate the change in the dependent variable associated with a one-unit change in the corresponding independent variable, while holding other variables constant.
The interpretation of coefficients depends on the specific context and scale of the variables. For example, in a simple linear regression with one independent variable, the coefficient represents the slope of the regression line, indicating the change in the dependent variable for each unit change in the independent variable.
Effect sizes, on the other hand, measure the magnitude of the relationship between variables. They provide a standardized measure that allows for comparisons across different studies or different variables. Effect sizes capture the strength or magnitude of the relationship, regardless of the scale of the variables.
Commonly used effect size measures in regression analysis include the coefficient of determination (R-squared), which represents the proportion of variance in the dependent variable explained by the independent variables, and the standardized regression coefficients (beta coefficients), which indicate the standardized change in the dependent variable for a one-standard-deviation change in the independent variable.
Do you do two sided hypothesis tests for regression?
Dependence on sample size: The p-value is influenced by the sample size. Large sample sizes tend to yield small p-values even for small effect sizes. This can result in statistically significant findings that may not have practical significance.
Multiple comparisons problem: Conducting multiple hypothesis tests increases the chance of obtaining false positives. When multiple tests are performed simultaneously, there is a higher likelihood of finding statistically significant results by chance alone. Adjustments such as the Bonferroni correction or controlling the family-wise error rate can be used to address this issue.
Misinterpretation and miscommunication: P-values are often misunderstood or misinterpreted, leading to incorrect conclusions or exaggerated claims. There is a risk of treating p-values as measures of truth or evidence, rather than as tools for quantifying the strength of evidence against the null hypothesis.
Let's explore the mathematical reasoning behind the influence of sample size on the p-value.
In hypothesis testing, the p-value is calculated as the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true. In many cases, the test statistic follows a known distribution under the null hypothesis.
For simplicity, let's consider a one-sample t-test where we are testing the mean of a population (null hypothesis) based on a sample mean. The test statistic in this case is the t-statistic. The formula to calculate the t-statistic is:
t = (x̄ - μ) / (s / √n)
where:
x̄ is the sample mean μ is the hypothesized population mean under the null hypothesis s is the sample standard deviation n is the sample size To calculate the p-value, we need to determine the probability of observing a t-statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true.
As the sample size (n) increases, the denominator of the t-statistic (s / √n) decreases. This means the t-statistic becomes larger for the same difference between the sample mean and the null hypothesis mean.
A larger t-statistic leads to a smaller p-value because the tails of the distribution (e.g., t-distribution) become narrower, and the probability of observing values as extreme as, or more extreme than, the observed t-statistic decreases. This is because larger sample sizes provide more precise estimates, reducing sampling variability and making it easier to detect small deviations from the null hypothesis.
In other words, with a larger sample size, it becomes easier to achieve statistical significance (i.e., small p-value) because the increased precision allows for a more accurate estimation of the population parameter.
In regression analysis, the influence of sample size on the p-values is similar to other hypothesis tests. However, there are a few specific considerations to keep in mind.
In linear regression, the p-values are typically associated with the significance of the regression coefficients (β coefficients) that represent the relationship between the predictor variables and the response variable.
When performing hypothesis tests on regression coefficients, such as t-tests or F-tests, the test statistics are calculated based on the estimated coefficients, their standard errors, and the distribution assumptions.
The t-test for a regression coefficient tests the null hypothesis that the true population coefficient is zero. The test statistic is calculated as:
t = (β - 0) / SE(β)
where:
β is the estimated coefficient SE(β) is the standard error of the coefficient The p-value associated with the t-test represents the probability of observing a t-statistic as extreme as, or more extreme than, the observed value under the assumption that the true coefficient is zero.
Similar to other hypothesis tests, as the sample size (n) increases in regression analysis, the standard errors tend to decrease. Smaller standard errors result in larger t-statistics, leading to smaller p-values.
With a larger sample size, the precision of the coefficient estimates improves, reducing sampling variability. This increased precision makes it easier to detect small effects, resulting in smaller p-values and potentially finding statistically significant relationships even for small effect sizes.
The formula for the standard error (SE) of a regression coefficient in linear regression depends on the specific regression model being used. In the case of simple linear regression (with one predictor variable), the formula for the standard error of the coefficient is:
SE(β) = sqrt(Σ(y - ŷ)^2 / ((n - 2) * Σ(x - x̄)^2))
where:
Σ represents the sum of the values
y is the observed response variable
ŷ is the predicted response variable based on the regression model
n is the sample size
x is the predictor variable
x̄ is the mean of the predictor variable
In multiple linear regression (with multiple predictor variables), the standard error of each coefficient is calculated based on the residual sum of squares (RSS) and the design matrix. The formula is more complex and involves matrix algebra. It can be represented as:
SE(β) = sqrt(diag(inv(X'X) * RSS / (n - k)))
where:
X is the design matrix containing the predictor variables
X' is the transpose of the design matrix
inv() represents the matrix inverse
diag() extracts the diagonal elements of a matrix
RSS is the residual sum of squares
n is the sample size
k is the number of predictor variables in the model
We have seen that test statistics are important for assessing the likelihood of observing coefficients as extreme or more than those in our sample data.
It appears that the probibility distributions which we use to compute a number that helps use form these conclusions around these test statistics
In statistics, degrees of freedom (df) refer to the number of values in a calculation that are free to vary. The concept of degrees of freedom is important in various statistical procedures and plays a role in determining the shape and characteristics of probability distributions, including the t-distribution.
Degrees of freedom are typically associated with the sample size and the number of restrictions or constraints in a statistical analysis. Let's explore their impact on distributions:
Sampling distributions: In inferential statistics, degrees of freedom are involved in calculating sampling distributions, which are used to make inferences about population parameters. The degrees of freedom are determined by the sample size and are often denoted as n - 1, where n represents the number of observations in the sample. For example, in a simple mean calculation, if you have a sample of size 10, the degrees of freedom would be 10 - 1 = 9.
t-distribution: The t-distribution is parameterized by its degrees of freedom. As the degrees of freedom increase, the shape of the t-distribution becomes closer to the standard normal distribution. Specifically, as the sample size increases, the t-distribution becomes more symmetrical and has thinner tails. For very large degrees of freedom (approaching infinity), the t-distribution converges to the standard normal distribution.
Chi-square distribution: The chi-square distribution, which is often used in hypothesis testing and confidence interval estimation, is also influenced by degrees of freedom. The degrees of freedom in the chi-square distribution determine the shape and spread of the distribution. As the degrees of freedom increase, the chi-square distribution becomes more bell-shaped and approaches a normal distribution.
F-distribution: The F-distribution, commonly used in analysis of variance (ANOVA) and regression analysis, is defined by two sets of degrees of freedom: the numerator degrees of freedom and the denominator degrees of freedom. The degrees of freedom in the numerator represent the number of groups or predictors, while the degrees of freedom in the denominator are related to the error or residual degrees of freedom. The F-distribution shape changes depending on the values of these degrees of freedom.
In summary, degrees of freedom in regression analysis reflect the balance between the number of observations, the number of predictors, and the number of estimated coefficients in the model. They help quantify the independent information available, the variability in the data, and are crucial for various statistical calculations and inference in regression analysis.
The F-distribution is a probability distribution that arises when testing hypotheses about variances or the ratio of variances. The F-test is commonly used in regression analysis to assess the overall significance of the regression model or to compare the significance of different models.
In regression, the F-test is used to test the null hypothesis that all the regression coefficients (except the intercept) are equal to zero. The F-score test statistic is calculated by dividing the mean square explained by the mean square unexplained:
F = (SSE / df_model) / (SSR / df_res)
where SSE represents the sum of squared errors (residual sum of squares), SSR represents the sum of squared regression (explained sum of squares), df_model represents the degrees of freedom associated with the regression model, and df_res represents the residual degrees of freedom.
The F-test statistic follows an F-distribution when the null hypothesis is true. The shape of the F-distribution depends on the degrees of freedom associated with the numerator (df_model) and the denominator (df_res). The F-distribution is positively skewed, with values ranging from zero to positive infinity.
By comparing the calculated F-test statistic to the critical value from the F-distribution corresponding to a chosen significance level, you can determine the statistical significance of the regression model. If the calculated F-value exceeds the critical value, you would reject the null hypothesis, indicating that the regression model as a whole has a significant effect.
In summary, the F-score test statistic in regression analysis is derived from the F-distribution. The F-distribution provides critical values for hypothesis testing, confidence intervals, and determining the significance of the overall model fit in regression analysis.
why are we using ratios in the numerator and denom of the F statistic. What does this accomplish mathmatically?
Using ratios in the numerator and denominator of the F statistic in regression analysis serves a specific purpose mathematically. Let's examine the rationale behind this approach:
Capturing variability: The F statistic is designed to capture the ratio of two sources of variability: the variability explained by the regression model and the variability that remains unexplained (residual variability). By comparing these two sources of variability, the F statistic allows us to assess whether the regression model as a whole explains a significant amount of the total variability in the data.
Comparing mean squares: The numerator of the F statistic represents the mean square explained, which is calculated by dividing the sum of squares explained (SSR) by the corresponding degrees of freedom (df_model). The denominator represents the mean square unexplained, calculated by dividing the sum of squares residual (SSE) by the residual degrees of freedom (df_res). Dividing these two mean squares provides a standardized measure of the difference between the explained and unexplained variability.
Focusing on variance ratios: Dividing the mean squares is mathematically equivalent to taking the ratio of the corresponding variances. Variance measures the dispersion or spread of a variable. By comparing the ratio of explained variance to unexplained variance, the F statistic allows us to assess whether the explained variance is significantly larger than the unexplained variance, indicating the presence of a significant relationship between the predictors and the response variable.
Utilizing F-distribution properties: The F statistic follows an F-distribution under the assumption of the null hypothesis being true. The F-distribution has two sets of degrees of freedom: the numerator degrees of freedom (associated with the explained variability) and the denominator degrees of freedom (associated with the unexplained variability). The shape of the F-distribution depends on these degrees of freedom, allowing us to determine critical values for hypothesis testing and assess the statistical significance of the model.
In summary, using ratios in the numerator and denominator of the F statistic helps to compare the explained and unexplained variability in regression analysis. This approach allows for standardized comparisons and leverages the properties of the F-distribution to assess the significance of the model.
Why are we dividing each by some degree of freedom?
Dividing the sum of squares by the corresponding degrees of freedom serves two primary purposes in regression analysis:
Standardization: Dividing the sum of squares by its associated degrees of freedom provides a standardized measure of the variability. The result is the mean square, which represents the average amount of variability per degree of freedom. Standardization allows for fair comparisons and facilitates the interpretation of the statistical quantities.
Scaling for degrees of freedom: The degrees of freedom reflect the amount of independent information available for estimating the variability in the data. By dividing the sum of squares by its associated degrees of freedom, we account for the available degrees of freedom and adjust the variability estimate accordingly. This scaling is important because as the degrees of freedom increase, the sum of squares tends to increase as well. Dividing by the degrees of freedom ensures that the mean square reflects the appropriate level of variability per degree of freedom.
In the context of the F statistic, dividing the sum of squares explained (SSR) by the degrees of freedom associated with the regression model (df_model) yields the mean square explained. Similarly, dividing the sum of squares residual (SSE) by the residual degrees of freedom (df_res) yields the mean square residual. These mean squares are then used to calculate the F statistic, which compares the ratio of explained variability to unexplained variability.
Dividing by the degrees of freedom is a way to normalize and adjust the variability measures in regression analysis, taking into account the available independent information and ensuring that the statistical quantities are appropriately scaled for comparison and interpretation.
Why are we dividing each by some degree of freedom?
Dividing the sum of squares by the corresponding degrees of freedom serves two primary purposes in regression analysis:
Standardization: Dividing the sum of squares by its associated degrees of freedom provides a standardized measure of the variability. The result is the mean square, which represents the average amount of variability per degree of freedom. Standardization allows for fair comparisons and facilitates the interpretation of the statistical quantities.
Scaling for degrees of freedom: The degrees of freedom reflect the amount of independent information available for estimating the variability in the data. By dividing the sum of squares by its associated degrees of freedom, we account for the available degrees of freedom and adjust the variability estimate accordingly. This scaling is important because as the degrees of freedom increase, the sum of squares tends to increase as well. Dividing by the degrees of freedom ensures that the mean square reflects the appropriate level of variability per degree of freedom.
In the context of the F statistic, dividing the sum of squares explained (SSR) by the degrees of freedom associated with the regression model (df_model) yields the mean square explained. Similarly, dividing the sum of squares residual (SSE) by the residual degrees of freedom (df_res) yields the mean square residual. These mean squares are then used to calculate the F statistic, which compares the ratio of explained variability to unexplained variability.
Dividing by the degrees of freedom is a way to normalize and adjust the variability measures in regression analysis, taking into account the available independent information and ensuring that the statistical quantities are appropriately scaled for comparison and interpretation.
In the context of the F statistic, the sum of squares regression (SSR) is the sum of the squared differences between the predicted values and the mean of the response variable. It quantifies the variability in the response variable that is explained by the regression model. On the other hand, the sum of squares residual (SSE) represents the sum of the squared differences between the observed response variable and the predicted response variable. It measures the unexplained variability in the response variable.
To clarify the correct formula:
SSR = Σ(ŷᵢ - ȳ)²
where:
SSR represents the sum of squares regression.
ŷᵢ is the predicted response variable (the value predicted by the regression model) for the ith observation.
ȳ is the mean of the observed response variable (the average value) for all observations.
Σ represents the summation symbol, indicating that you need to sum up the squared differences across all observations.
In the F statistic, dividing the SSR by the degrees of freedom associated with the regression model (df_model) yields the mean square regression, denoted as MSR. Dividing the sum of squares residual (SSE) by the residual degrees of freedom (df_res) yields the mean square residual, denoted as MSE.
The F statistic is then calculated as the ratio of MSR to MSE:
F = MSR / MSE
Are F tests statistics calculated per coefficient or how is this done on a coefficient by coefficient basis?
While the F-test assesses the overall significance of the regression model, if you are interested in determining the significance of individual coefficients, you would typically look at the corresponding t-tests or calculate confidence intervals for each coefficient separately.
how is the t statistic computed for coefficients?
The t-statistic is computed for coefficients in regression analysis to assess the significance of individual coefficients. The t-statistic measures the number of standard errors that a coefficient estimate is away from zero. It is calculated by dividing the estimated coefficient value by its standard error.
The formula for calculating the t-statistic for a coefficient estimate is as follows:
t = (coefficient estimate) / (standard error of coefficient)
where:
The coefficient estimate is the estimated value of the coefficient obtained from the regression analysis. The standard error of the coefficient is a measure of the uncertainty or variability associated with the coefficient estimate. The t-statistic follows a t-distribution with a certain number of degrees of freedom, typically based on the residual degrees of freedom in the regression model. By comparing the calculated t-statistic to the critical values from the t-distribution for a given significance level, you can determine whether the coefficient is statistically significant or not. If the absolute value of the t-statistic is greater than the critical value, it suggests that the coefficient is statistically significant, indicating that it has a non-zero effect on the response variable.
In summary, the t-statistic is computed by dividing the coefficient estimate by its standard error and is used to assess the significance of individual coefficients in regression analysis.
What is the formula for the standard error of a coefficient? The formula for the standard error of a coefficient estimate in regression analysis is as follows:
Standard Error = sqrt(MSE * (X'X)^(-1) )
where:
MSE represents the mean square residual, which is the sum of squares residual divided by the residual degrees of freedom.
X is the design matrix that includes the predictor variables.
X' represents the transpose of the design matrix.
(X'X)^(-1) represents the inverse of the matrix product of the transpose of X and X.
In other words, the standard error of a coefficient estimate is calculated by taking the square root of the mean square residual multiplied by the inverse of the matrix product of the transpose of the design matrix and the design matrix itself.
The standard error of a coefficient estimate measures the variability or uncertainty associated with the estimated coefficient. It represents the average amount by which the estimated coefficient would vary across different samples taken from the same population.
The standard error is used to calculate t-statistics and construct confidence intervals for coefficient estimates, which are important in assessing the statistical significance and precision of the estimated coefficients in the regression model.
what information does the design matrix contain?
The design matrix, denoted as X, contains the predictor variables used in a regression model. Each row of the design matrix represents an observation or data point, and each column represents a predictor variable or a constant term.
The design matrix contains the following information:
Predictor Variables: Each column of the design matrix represents a predictor variable. These variables can be continuous, categorical, or a combination of both. Continuous variables capture numerical data, while categorical variables represent groups or categories.
Constant Term: The design matrix typically includes a column of ones (or a constant term) to capture the intercept or the baseline value in the regression model. This column ensures that the regression line or hyperplane is not constrained to pass through the origin.
Transformations and Interactions: The design matrix can also include transformed variables or interaction terms that capture nonlinear relationships or the combined effect of multiple predictors.
By organizing the predictor variables into a design matrix, you can easily represent the relationship between the predictors and the response variable in a linear equation or matrix form. The design matrix serves as the input for estimating the regression coefficients and performing various calculations in regression analysis, such as computing the standard error, conducting hypothesis tests, and making predictions.
t = (coefficient estimate - hypothesized value) / (standard error of coefficient)
The mean square error (MSE) is a measure of the average squared difference between the observed values and the predicted values in regression analysis. It quantifies the overall quality of the regression model's predictions.
To compute the MSE, you typically follow these steps:
Start with a set of observed values (y) and the corresponding predicted values (ŷ) from your regression model.
Calculate the residual for each observation, which is the difference between the observed value and the predicted value: Residual = y - ŷ.
Square each residual to eliminate negative values and emphasize larger errors: Squared Residual = (y - ŷ)^2.
Sum up all the squared residuals: Sum of Squared Residuals = Σ(y - ŷ)^2.
Divide the sum of squared residuals by the total number of observations (n) to get the mean squared error: MSE = (1/n) * Σ(y - ŷ)^2.
The MSE represents the average of the squared differences between the observed values and the predicted values. It is a measure of how well the regression model fits the data, with smaller values indicating better fit and lower prediction errors.
Note that the MSE is specific to a particular regression model and is affected by the choice of predictor variables, model assumptions, and the quality of the data. It is commonly used in evaluating and comparing different regression models based on their prediction accuracy.
The degrees of freedom for the numerator (df₁) in the F-statistic correspond to the number of predictors or independent variables in the model. It represents the number of parameters being estimated.
The degrees of freedom for the denominator (df₂) in the F-statistic correspond to the residual degrees of freedom, which are the total number of observations minus the number of parameters estimated in the model.
Degrees of freedom cont.
that you're estimating a term (variance) that itself depends on an estimate (the mean). The mean is not known, but has error to it. By using N instead of N-1, you are not accounting for the unique bit of information that gives you the mean estimate. I.e., if you have an N of 10, then your mean estimate is uniquely determined by one of those values - You can sum 9 of those scores, and the tenth is what determines your mean. So you only actually have 9 unique pieces of information about your mean; the last one determines your mean, and your mean itself is an estimate of a population mean, with error to it. That 'last' value is ultimately what produces error in the mean estimate, so to speak. So instead of thinking of having N data points, think about having N-1 data points for the mean, and one data point determining the error of the estimated mean from the true mean.
--Reddit user StephenSRMMartin
GLM stands for Generalized Linear Model. It is called "generalized" because it extends the concept of linear regression to handle a broader range of response variables that may not follow a normal distribution. While linear regression is suitable for continuous outcomes, GLM allows for modeling other types of responses such as binary, count, or categorical variables.
GLM incorporates a link function that connects the linear predictor (a linear combination of the predictors) to the response variable. This link function provides flexibility in modeling the relationship between the predictors and the response, allowing for non-linear associations. Additionally, GLM allows for the specification of a probability distribution from the exponential family that best represents the response variable's characteristics.
By accommodating different types of response variables and allowing for non-linear relationships, GLM provides a powerful framework for analyzing a wide range of data in a unified manner.
Moving from multiple regression to general linear models (GLMs) may be necessary in several situations:
Nonlinear relationships: If the relationship between the dependent variable and predictors is nonlinear, GLMs allow you to incorporate nonlinear terms (such as quadratic or interaction terms) to capture more complex patterns.
Non-continuous outcomes: Multiple regression assumes a continuous dependent variable, but GLMs can handle various types of outcomes, including binary (e.g., logistic regression), count (e.g., Poisson regression), and categorical (e.g., multinomial regression).
Violations of assumptions: If the assumptions of multiple regression are violated, such as non-constant variance (heteroscedasticity) or non-normality of residuals, GLMs provide alternative models that can handle these violations. For example, generalized least squares (GLS) can handle heteroscedasticity, and generalized estimating equations (GEE) can address correlated errors.
Censored or truncated data: GLMs, such as censored regression models (e.g., Tobit regression) or survival analysis (e.g., Cox proportional hazards model), can handle data with censoring or truncation, where the outcome is not fully observed for all cases.
Nested or clustered data: If your data have a hierarchical structure, such as individuals nested within groups or repeated measures on the same individuals, GLMs with random effects (such as mixed-effects models) can account for the dependencies in the data and provide more accurate estimates.
Non-constant error variance: GLMs can handle situations where the assumption of constant error variance is violated, such as when the variability of the residuals changes across levels of predictors (e.g., by using weighted least squares or robust regression).
Give me an example where I have categorical and continuous predictors where regression falls short but GLMs with random effects can help
Let's consider an example where we want to predict the sales of a product based on both categorical and continuous predictors. In this scenario, we have data on sales (continuous outcome variable), advertising expenditure (continuous predictor), product category (categorical predictor), and store location (categorical predictor).
In multiple regression, we could include the continuous predictor (advertising expenditure) and dummy variables for the categorical predictors (product category and store location) as independent variables. However, multiple regression assumes that the residuals are independent and have constant variance, which may not hold if there are underlying hierarchical structures in the data.
To address this limitation, we can use a GLM with random effects. Here, we can include random intercepts for each store location to account for the potential clustering of sales within stores. This random intercept captures the variability across stores that is not explained by the fixed effects (such as advertising expenditure and product category). By including the random effects, we allow for the correlation among observations within the same store, which helps to provide more accurate estimates and account for potential heterogeneity across stores.
The random effects in the GLM capture the additional sources of variation beyond the fixed effects and can help to improve the model's fit and predictive accuracy. The model takes into account the hierarchical structure of the data, where sales are nested within store locations. This approach allows for the estimation of both fixed effects (such as the effect of advertising expenditure) and random effects (variation across store locations).
In summary, when dealing with data that have both categorical and continuous predictors and potential hierarchical structures, multiple regression may fall short in capturing the full complexity of the data. In such cases, a GLM with random effects provides a more appropriate modeling framework to account for the dependencies and heterogeneity within the data.
Now use the same example without the need for random effects. Tell me why GLM is useful based on the mixture of continuous and categorical predictors
In the example without the need for random effects, the use of Generalized Linear Models (GLMs) can still be beneficial when dealing with a mixture of continuous and categorical predictors. Here's why:
Flexibility in outcome distribution: GLMs allow for modeling the relationship between the predictors and the outcome variable even when the distribution of the outcome is not necessarily normally distributed. This flexibility is particularly useful when the outcome variable is binary (e.g., success/failure), count-based (e.g., number of events), or follows a different non-normal distribution.
Categorical predictor handling: GLMs provide a straightforward way to incorporate categorical predictors in the analysis. By creating appropriate dummy variables or using suitable coding schemes, categorical predictors can be included in the model, allowing for the estimation of their effects on the outcome variable. This makes it easier to examine the relationship between the categorical predictors and the outcome while properly accounting for their non-continuous nature.
Nonlinear relationships: GLMs can capture nonlinear relationships between the predictors and the outcome by using appropriate link functions. For example, if the relationship between the predictors and the outcome is expected to be exponential (e.g., in Poisson regression), GLMs with an appropriate exponential link function can accommodate this nonlinearity.
Handling overdispersion: In some cases, the variability in the outcome variable may exceed what is expected under the assumed distribution. GLMs allow for addressing overdispersion, where the variance of the outcome variable is greater than the mean, by incorporating dispersion parameters or using alternative distributional assumptions (e.g., negative binomial regression).
Hypothesis testing and inference: GLMs provide tools for hypothesis testing and inference, similar to multiple regression. Coefficient estimates, standard errors, p-values, and confidence intervals can be obtained to assess the statistical significance and precision of the estimated effects of the predictors.
Give me a fundamental tenent for thinking about bias in any estimator and how degrees of freedom fits into understanding bias? A fundamental tenet for thinking about bias in any estimator is that bias refers to the systematic deviation of the estimator's expected value from the true value of the parameter being estimated. In other words, bias measures the tendency of an estimator to consistently overestimate or underestimate the true value.
Degrees of freedom play a role in understanding bias because they affect the precision and reliability of an estimator. In statistical inference, degrees of freedom represent the number of independent pieces of information available for estimating a parameter or making inferences about a population. A higher number of degrees of freedom generally indicates a larger sample size or more information available for estimation.
When the degrees of freedom are too low, the estimator can be biased due to insufficient information. This can happen when the sample size is small or when there is high collinearity among the predictors in a regression model. In such cases, the estimator may have a tendency to be more influenced by the specific characteristics of the sample, leading to biased estimates of the parameters.
On the other hand, as the degrees of freedom increase, the estimator becomes less biased and more efficient. With a larger sample size or more independent information, the estimator can better capture the underlying population characteristics and reduce the influence of random variation in the sample.
Therefore, understanding the degrees of freedom is important in assessing the bias of an estimator and the precision of the estimated parameters. It helps to consider the sample size, the number of predictors, and the potential sources of dependence or collinearity in the data to ensure reliable and unbiased estimation.
User what are measures of collinearity ChatGPT Measures of collinearity are used to assess the degree of linear relationship between predictor variables in a regression model. High collinearity among predictors can lead to instability and unreliable estimates of the regression coefficients. Here are some common measures of collinearity:
Correlation coefficient (Pearson's correlation): This measure quantifies the linear relationship between two variables and ranges from -1 to 1. A correlation coefficient close to -1 or 1 indicates a strong linear relationship, while a value close to 0 suggests a weak or no relationship.
Variance Inflation Factor (VIF): VIF measures the extent to which the variance of the estimated regression coefficient is inflated due to collinearity. It assesses how much the variance of a coefficient is increased compared to when there is no collinearity. A VIF value greater than 1 suggests some level of collinearity, with higher values indicating stronger collinearity. Typically, a VIF value above 5 or 10 is considered indicative of high collinearity.
Tolerance: Tolerance is the reciprocal of VIF (1/VIF). It indicates the proportion of variance in a predictor that is not explained by other predictors in the model. A tolerance value close to 1 suggests low collinearity, while a value close to 0 indicates high collinearity.
Condition Number: The condition number is a measure of the overall collinearity in the regression model. It is calculated as the square root of the ratio of the largest eigenvalue to the smallest eigenvalue of the correlation matrix of the predictors. A condition number greater than 30 or 100 indicates substantial collinearity.
These measures provide insights into the presence and severity of collinearity among predictors. If high collinearity is detected, it may be necessary to take corrective actions such as removing redundant predictors, transforming variables, or considering dimensionality reduction techniques to improve the stability and interpretability of the regression model.