Testing of Hypothesis

Testing of Hypothesis, Hypothesis testing, Independent t test, Independent z test, Analysis of variance, ANOVA, Comparison tests

Measure of Association: Contingency Table (2019)

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The Contingency Table (also called two-way frequency tables/ crosstabs or cross-tabulations) is used to find the relationship (association or dependencies (a measure of association)) between two or more variables measured on the nominal or ordinal measurement scale.

Contingency Table: A Measure of Association

A contingency table contains $R$ rows and $C$ columns measured, the order of the contingency table is $R \times C$. There should be a minimum of 2 (categories in row variable without row header) and 2 (categories in column variable without column header).

A cross table is created by listing all the categories (groups or levels) of one variable as rows in the table and the categories (groups or levels) of other (second) variables as columns, and then joint (cell) frequency (or counts) for each cell. The cell frequencies are totaled across both the rows and the columns. These totals (sums) are called marginal frequencies. The sum (total) of column sums (or rows sum) can be called the Grand Total and must be equal to $N$. The frequencies or counts in each cell are the observed frequency.

The next step in calculating the Chi-square statistics is the computation of the expected frequency for each cell of the contingency table. The expected values of each cell are computed by multiplying the marginal frequencies of the row and marginal frequencies of the column (row sums and column sums are multiplied) and then dividing by the total number of observations (Grand Total, $N$). It can be formulated as

$Expected\,\, Frequency = \frac{(Row\,\, Total \,\, * \,\, Column\,\, Total)}{ Grand \,\, Total}$

The same procedure is used to compute the expected frequencies for all the cells of the contingency table.

The next step is related to the computation of the amount of deviation or error for each cell. for this purpose subtract the expected cell frequency from the observed cell frequency for each cell. The Chi-square statistic is computed by squaring the difference and then dividing the square of the difference by the expected frequency for each cell.

Contingency Table Measure of Association

Finally, the aggregate Chi-square statistic is computed by summing the Chi-square statistic. For formula is,

$$\chi^2=\sum_{i=1}^n \frac{\left(O_{if}-E_{ij}\right)^2}{E_{ij}}$$

The $\chi^2$ table value, the degrees of freedom, and the level of significance are required. The degrees of freedom for a contingency table is computed as
$$df=(number\,\, of \,\, rows – 1)(number \,\, of \,\, columns -1)$$.

For further detail about the contingency table (as a measure of association) and its example about how to compute expected frequencies and Chi-Square statistics, see the video lecture

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Student t-test Comparison Test (2015)

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In 1908, William Sealy Gosset published his work under the pseudonym “Student” to solve problems associated with inference based on sample(s) drawn from a normally distributed population when the population standard deviation is unknown. He developed the Student t-test and t-distribution, which can be used to compare two small sets of quantitative data collected independently of one another, in this case, this t-test is called independent samples t-test or also called unpaired samples t-test.

The Student t-test is the most commonly used statistical technique in testing of hypothesis based on the difference between sample means. The student t-test can be computed just by knowing the means, standard deviations, and number of data points in both samples by using the following formula

\[t=\frac{\overline{X}_1-\overline{X}_2 }{\sqrt{s_p^2 (\frac{1}{n_1}+\frac{1}{n_2})}}\]

where $s_p^2$ is the pooled (combined) variance and can be computed as

\[s_p^2=\frac{(n_1-1)s_1^2 + (n_2-2)s_2^2}{n_1+n_2-2}\]

Using this test statistic, we test the null hypothesis $H_0:\mu_1=\mu_2$ which means that both samples came from the same population under the given “level of significance” or “level of risk”.

If the computed t-statistics from the above formula is greater than the critical value (value from t-table with $n_1+n_2-2$ degrees of freedom and given a level of significance, say $\alpha=0.05$), the null hypothesis will be rejected, otherwise, the null hypothesis will be accepted.

Note that the t-distribution is a family of curves depending on the degree of freedom (the number of independent observations in the sample minus the number of parameters). As the sample size increases, the t-distribution approaches a bell shape i.e. normal distribution.

Student t-test Example

The production manager wants to compare the number of defective products produced on the day shift with the number on the afternoon shift. A sample of the production from 6-day and 8-afternoon shifts revealed the following defects. The production manager wants to check at the 0.05 significance level, is there a significant difference in the mean number of defects per shits?

Day shift587697  
Afternoon Shit810711912149

Some required calculations for the Student t-test are:

The mean of samples:

$\overline{X}_1=7$, $\overline{X}_2=10$,

Standard Deviation of samples

$s_1=1.4142$, $s_2=2.2678$ and $s_p^2=\frac{(6-1) (1.4142)^2+(8-1)(2.2678)^2}{6+8-2}=3.8333$

Step 1: Null and alternative hypothesis are: $H_0:\mu_1=\mu_2$ vs $H_1:\mu_1 \ne \mu_2$

Step 2: Level of significance: $\alpha=0.05$

Step 3: Test Statistics

$\begin{aligned}
t&=\frac{\overline{X}_1-\overline{X}_2 }{\sqrt{s_p^2 (\frac{1}{n_1}+\frac{1}{n_2})}}\\
&=\frac{7-10}{\sqrt{3.8333(\frac{1}{6}+\frac{1}{8})}}=-2.837
\end{aligned}$

Step 4: Critical value or rejection region (Reject $H_0$ if the absolute value of t-calculated in step 3 is greater than the absolute table value i.e. $|t_{calculated}|\ge t_{tabulated}|$). In this example t-tabulated is -2.179 with 12 degrees of freedom at a significance level of 5%.

Step 5: Conclusion: As computed value $|2.837| > |2.179|$, the number of defects is not the same on the two shifts.

Different Types of Comparison Tests

  • Independent Samples t-test: This compares the means of two independent groups. For example, you might use this to see if a new fertilizer increases plant growth compared to a control group.
  • Paired Samples t-test: This compares the means from the same group at different times or under various conditions. Imagine testing the same group’s performance on a task before and after training.
  • One-Sample t-test: This compares the mean of a single group to a hypothesized value. For instance, you could use this to see if students’ average exam scores significantly differ from 75%.

The summary of key differences between the comparison tests

Independent SamplesPaired SamplesOne-Sample
GroupsIndependentSame group at different timesSingle group
HypothesisMeans are differentMeans are differentMean is different from a hypothesized value
AssumptionsNormally distributed data, equal variances (testable)Normally distributed differencesNormally distributed data

Regardless of the type of t-test, all the above comparison tests assess the significance of a difference between means. These tests tell the research if the observed difference is likely due to random chance or reflects a true underlying difference in the populations.

Student T-test

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P-value Definition, Interpretation, Introduction, Significance

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In this post, we will discuss the P-value definition, interpretation, introduction, and some related examples.

P-value Definition

The P-value also known as the observed level of significance or exact level of significance or the exact probability of committing a type-I error (probability of rejecting $H_0$, when it is true), helps to determine the significance of results from the hypothesis. The P-value is the probability of obtaining the observed sample results or a more extreme result when the null hypothesis (a statement about population) is true.

In technical words, one can define the P-value as the lowest level of significance at which a null hypothesis can be rejected. If the P-value is very small or less than the threshold value (chosen level of significance), then the observed data is considered inconsistent with the assumption that the null hypothesis is true, and thus null hypothesis must be rejected while the alternative hypothesis should be accepted. A P-value is a number between 0 and 1 in literature.

Usual P-value Interpretation

  • A small P-value (<0.05) indicates strong evidence against the null hypothesis
  • A large P-value (>0.05) indicates weak evidence against the null hypothesis.
  • p-value very close to the cutoff (say 0.05) is considered to be marginal.

Let the P-value of a certain test statistic is 0.002 then it means that the probability of committing a type-I error (making a wrong decision) is about 0.2 percent, which is only about 2 in 1,000. For a given sample size, as | t | (or any test statistic) increases the P-value decreases, so one can reject the null hypothesis with increasing confidence.

p value and significance level

Fixing the significance level ($\alpha$, i.e. type-I error) equal to the p-value of a test statistic then there is no conflict between the two values, in other words, it is better to give up fixing up (significance level) arbitrary at some level of significance such as (5%, 10%, etc.) and simply choose the P-value of the test statistic. For example, if the p-value of the test statistic is about 0.145 then one can reject the null hypothesis at this exact significance level as nothing wrong with taking a chance of being wrong 14.5% of the time someone rejects the null hypothesis.

P-value addresses only one question: how likely are your data, assuming a true null hypothesis? It does not measure support for the alternative hypothesis.

Most authors refer to a P-value<0.05 as statistically significant and a P-value<0.001 as highly statistically significant (less than one in a thousand chance of being wrong).

P-value Definition, P-value Interpretation

The P-value interpretation is usually incorrect as it is usually interpreted as the probability of making a mistake by rejecting a true null hypothesis (a Type-I error). The P-value cannot be the error rate because:

The P-value is calculated based on the assumption that the null hypothesis is true and that the difference in the sample is by random chance. Consequently, a p-value cannot tell about the probability that the null hypothesis is true or false because it is 100% true from the perspective of the calculations.

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