Hotelling's T-square statistic

Examples of Hotelling's T-square statistic in the following topics:

• Multivariate Testing

  • Hotelling's $T$-square statistic allows for the testing of hypotheses on multiple (often correlated) measures within the same sample.
  • A generalization of Student's $t$-statistic, called Hotelling's $T$-square statistic, allows for the testing of hypotheses on multiple (often correlated) measures within the same sample.
  • Hotelling's $T^2$ statistic follows a $T^2$ distribution.
  • Hotelling's $T$-squared distribution is important because it arises as the distribution of a set of statistics which are natural generalizations of the statistics underlying Student's $t$-distribution.
  • The test statistic is defined as:

• References

  • ., Bradley, T.
  • D. (1979) Type I error rate of the chi square test of independence in r x c tables that have small expected frequencies.

• The Root-Mean-Square

  • The root-mean-square, also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity, or set of numbers.
  • The root-mean-square, also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity, or set of numbers.
  • Its name comes from its definition as the square root of the mean of the squares of the values.
  • Computing the average of this set of numbers wouldn't tell us much because the negative numbers cancel out the positive numbers, resulting in an average of zero.
  • Physical scientists often use the term "root-mean-square" as a synonym for standard deviation when referring to the square root of the mean squared deviation of a signal from a given baseline or fit.

• Structure of the Chi-Squared Test

  • The chi-square ($\chi^2$) test is a nonparametric statistical technique used to determine if a distribution of observed frequencies differs from the theoretical expected frequencies.
  • Chi-square statistics use nominal (categorical) or ordinal level data.
  • First, we calculate a chi-square test statistic.
  • We may observe data that give us a high test-statistic just by chance, but the chi-square distribution shows us how likely it is.
  • This is a property shared by the $T$-distribution.

• Root Mean Square Values

  • The root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity.
  • The corresponding formula for a continuous function f(t) defined over the interval T1 ≤ t ≤ T2 is as follows:
  • V is the voltage at time t, V0 is the peak voltage, and f is the frequency in hertz.
  • Here, I is the current at time t, and I0=V0/R is the peak current.
  • Since V0 is a constant, we can factor it out of the square root, and use a trig identity to replace the squared sine function.

• Quantitative or Qualitative Data?

  • Paired and unpaired t-tests and z-tests are just some of the statistical tests that can be used to test quantitative data.
  • A t-test is any statistical hypothesis test in which the test statistic follows a t distribution if the null hypothesis is supported.
  • When the scaling term is unknown and is replaced by an estimate based on the data, the test statistic (under certain conditions) follows a t distribution .
  • One of the most common statistical tests for qualitative data is the chi-square test (both the goodness of fit test and test of independence).
  • Plots of the t distribution for several different degrees of freedom.

• Calculations for the t-Test: One Sample

  • In each case, the formula for a test statistic that either exactly follows or closely approximates a $t$-distribution under the null hypothesis is given.
  • Each of these statistics can be used to carry out either a one-tailed test or a two-tailed test.
  • Once a $t$-value is determined, a $p$-value can be found using a table of values from Student's $t$-distribution.
  • Let $\hat{\alpha}$ and $\hat{\beta}$ be least-squares estimators, and let $SE_\hat{\alpha}$ and $SE_\hat{\beta}$, respectively, be the standard errors of those least-squares estimators.
  • Therefore, the sum of the squares of residuals, or $SSR$, is given by:

• Understanding regression output from software

  • We will generally label the test statistic using a T, since it follows the t distribution.
  • We previously used a t test statistic for hypothesis testing in the context of numerical data.
  • Table 7.21 offers the degrees of freedom for the test statistic T: df = 25.
  • We could have identified the t test statistic from the software output in Table 7.21, shown in the second row (unemp) and third column (t value).
  • Table 7.23 shows statistical software output from fitting the least squares regression line shown in Figure 7.16.

• Example: Test for Independence

  • The chi-square test for independence is used to determine the relationship between two variables of a sample.
  • To examine statistically whether boys got in trouble more often in school, we need to establish hypotheses for the question.
  • where $\sigma_r$ is the sum over that row, $\sigma_c$ is the sum over that column, and $\sigma_t$ is the sum over the entire table.
  • With the values in the table, the chi-square statistic can be calculated as follows:
  • In the chi-square test for independence, the degrees of freedom are found as follows:

• Finding the least squares line

  • A common exercise to become more familiar with foundations of least squares regression is to use basic summary statistics and point-slope form to produce the least squares line.
  • The third column is a t test statistic for the null hypothesis that β1 = 0: T = −3.98.
  • The last column is the p-value for the t test statistic for the null hypothesis β1 = 0 and a two-sided alternative hypothesis: 0.0002.
  • Summary of least squares fit for the Elmhurst data.
  • In the left panel, a straight line does not fit the data.