The world of mathematics often thrives on assumptions that help support theories and make calculations more convenient. While statistics in itself is a huge concept and can be very confusing frequently, two concepts also operate on such assumptions and they are t-tests and p-values.
T-Test vs P-Value
The main difference between a t-test and a p-value lies in the quantity they measure. A t-test measures the rate of difference in the population whereas a p-value measures the probability of attaining a t-test value that is at least as big as the value attained in the sample data. This can also be explained as the strength of proof that the result obtained is not just a likely chance occurrence.
Difference Between T-Test and P-Value in Tabular Form
|Parameters of Comparison||T-Test||P-Value|
|Definition||t-test can be explained as a method of statistics that helps determine whether there is a significant difference between the two sets of data recorded.||the p-value can be defined as the probability of obtaining a test result that is at least as big as the actual observed result when the null hypothesis stands true.|
|Terminology||The term t-test stands for ‘test statistic’.||The term p-value stands for ‘probability value’.|
|Values required||We need the mean values of each data set, a standard deviation of every group, and the number of data samples in each group.||We need the deviation of the value observed from the standard value chosen while the probability distribution of the statistic is given.|
|Scope||The corresponding p-value of a t-test can be calculated by the use of a t-distribution table or an online calculator or software.||The corresponding t-test value of a p-value can be calculated by the use of an inverse Cumulative Distribution Function(CDF).|
|Averages||The averages of samples in a t-test are alternating.||The averages of samples while calculating the p-value are null-same.|
|Results obtained||We obtain the difference between the two mean values.||We obtain a conclusion as to whether there is enough evidence to reject the null hypothesis.|
What is a T-Test?
A T-test is defined as an inferential statistical method that allows us to compare the mean values of two different sets and determine if they originate from the same kind of population. However, while making straightforward calculations, the results can turn out to be quite ambiguous. Thus, mathematicians all over the world make certain assumptions to make calculations easier and more accurate. Let us look at some of them:
Few Assumptions Made During T-Tests
- The first assumption made during a t-test is referring to the data collected from a random sample. Any data collected from a certain representative is said to be that of a portion that is selected randomly.
- The second assumption made is related to the samples collected. Any observations collected in a sample will remain independent of any other observation in other samples.
- The third assumption relates to the scale of measurement. It says that the scale applied to the collected data will follow either a continuous or an ordinal scale.
- The fourth assumption is made concerning the plots of data. The data when plotted will produce a normal distribution and result in a bell-shaped distribution curve.
- The fifth assumption is related to homogeneity. When the resultant standard deviations of the collected samples are almost equal, it results in equal variance. Thus, we acquire homogeneity of variance.
Ways to Calculate a T-Test
As previously mentioned, we need 3 proper values to calculate the t-value they are mean values from each of the data sets, the standard deviation obtained from each group, and the number of values collected in each group. The calculation of a t-test produces the t-value that will further be compared against the respective value in the distribution table to check whether the difference between the means is outside the possible range. This means with the help of a t-test, we can figure out whether the difference is significant i.e., true difference or a random, meaningless difference.
A t-distribution table can be described as a method of presenting data that, when plotted, falls in the bell curve. T-distribution can be explained as a type of normal distribution that is usually used for small sample sizes. It is also referred to as the student’s t-distribution. In a t-distribution, there are higher numbers of observations close to the mean and fewer observations towards the tails. There are two types of distribution tables: one-tailed and two-tailed. The one-tailed format is used to analyze cases that establish a fixed value or a fixed range with clarity in the sign of the value i.e., a positive value or a negative value. On the other hand, two-tailed formats are used to assess random bound analysis, which means checking if the coordinates fall within a certain range.
The variance in a t-distribution can be calculated using the number of degrees of freedom in a data set. However, it is a more conservative form of the standard normal distribution which is also referred to as the ‘Z-distribution’.
Through the computation of a t-test, we acquire two values and they are t-values and degrees of freedom. A t-value can be described as the ratio of the difference between the two means of the sample sets to the variation between the sample sets. While the value of the numerator can be pretty easy to calculate, the value in the denominator can sometimes be miscalculated. We can infer from the formula that the smaller the t value, the more identical the two sample sets are. Similarly, we can say that the higher the value of the t-value, the high is the difference that exists between the two sets. The t-values can also be called t-scores. Therefore,
- Large t-score means that the given groups are different.
- Small t-scores mean that the given groups are somewhat similar.
Degrees of Freedom
Degrees of freedom can be explained as the number of values in a certain calculation that is allowed to vary. They are necessary for analyzing the significance and the requirement of the null hypothesis. The degrees of freedom for a given data set can be calculated as the total number of observations decreased by 1.
With an increase in the number of degrees of freedom, the curve of t-distribution will get closer to the curve of the standard normal distribution i.e., the z-distribution curve until the curves look almost identical. Through several data sets, we can infer and approximate that above 30 degrees of freedom, the curve of t-distribution will match the curve of z-distribution. This means for data sets with large sample sizes, we can replace the t-distribution by using the z-distribution.
Types of T-Tests
- Paired T-Test: If the samples obtained contain matching pairs of similar units, we can use the paired t-test. This can also be done when there are recurring values of the same measure. This type of test can be referred to as a correlated t-test. Eg:- When the pre-treatment and post-treatment results of a patient who’s being analyzed frequently, the patient’s sample will be used against their sample as a control sample.
Where m1= mean of the first sample set
m2= mean of the second sample set
S.D= Standard deviation(S.D) of the difference between the paired data values.
n= sample size, also defined as the number of paired differences
n-1=degrees of freedom for that sample set.
- Pooled T-Test: this type of t-test can be used when there is an equal number of samples used in each group. This can also be used when there is homogeneity in the variance of the two groups. We also have formulas to calculate the t-values and degrees of freedom while calculating through a pooled t-test.
Where m1= mean of the first sample set
m2= mean of the second sample set
n1=number of data values in the first sample set
n2= number of data values in the second sample set.
v1=variance of the first sample set.
v2=variance of the second sample set.
- Welch’s T-Test: Also known as the unequal variance t-test, this type of t-test is employed when the number of samples in both the data sets is different. Thus there will be no homogeneity in the variance between the two sets. The t-value and the degrees of freedom of the datasets while employing the use of welch’s set is:
Degrees of freedom= (v1n1+v2n2)2(v12/n1)2n1-1+(v22/n2)2n2-1
The terminology is the same as explained above.
What is P-Value?
How to Calculate a P-Value
P-values can be calculated using online calculators or statistical programs such as R or SPSS. As stated previously, the p-value can be calculated based on the test statistic and the degrees of freedom in that particular data set. We can say that the calculation of a p-value differs based on:
- The kind of statistical test being employed to evaluate the hypothesis since the assumptions differ from method to method. Three types of tests will describe the location on the probability distribution curve. The tests are the upper-tailed test, lower-tailed test, and the two-sided test.
- Number of independent variables can directly affect the number of degrees of freedom and how large or small the test statistic must be to generate the same p-value.
Important Points to Note When Representing a P-Value
- P-values are generally represented until the second or the third decimal place.
- It is more accurate when ‘0’ is not used in front of the decimal point. Eg:- we shouldn’t write p=0.005 but instead, p= .005 as it is more accurate.
- The term ‘p’ must always be italicized and be represented as ‘p’.
- p=.000 is not accurate as some statistical packages such as SPSS can sometimes output. Thus, it must be represented as p<0.001 , as much as possible.
- When attempting to use the most accurate terminology, we must say that the contradiction of significance can be termed as insignificant and not non-significant.
Statistical Significance of a P-Value
As mentioned previously, the p-value describes the probability of the occurrence of data by a random chance, which means evaluating the hypothesis of the null hypothesis. The value of the p-value ranges anywhere from 0 to 1. Through general calculations, we can infer that as the p-value gets smaller, the evidence of the null hypothesis being invalid gets stronger.
- p≤0.05: When the p-value of a certain data set is lesser than 0.05, it is considered statistically insignificant. The mathematical representation of the same depicts that there is less than a 5% probability of the null hypothesis is valid. Thus, when we summarize that the null hypothesis must be rejected and accept the alternative hypothesis. However, we need to note that we cannot consider the alternative theory to be true by a 95% percent probability. However relevant the p-value can be to the null hypothesis, it does not evaluate the validity of the alternative hypothesis.
- p≥0.05: When the p-value of a data set is higher than 0.05, it is not considered statistically insignificant. Unlike the above case, where there is strong evidence supporting the validation of the null hypothesis. Thus, we can reject the alternative hypothesis and retain the use of the null hypothesis. However, by ‘rejecting’ the hypothesis for a case, we do not exactly debunk the hypothesis itself but instead, evaluate whether the obtained results support the hypothesis or do not support the hypothesis.
Main Differences Between T-Test and P-Value In Points
- The t-test can be explained as a method of inferential statistics that evaluates whether there is a significant difference between the two data sets recorded. On the other hand, the p-value can be defined as the probability of obtaining an absolute t-value, considering the null hypothesis stands valid after evaluation.
- The term t-test stands for 'test statistic' whereas the term p-value stands for 'probability value'.
- The values required to calculate a t-value are the mean values of each data set, the standard deviation of each group, and the number of data samples recorded in each group. On the other hand, we need the value of the deviation of the observed value from the standard value.
- Through a t-test, we can obtain the t-values and the number of degrees of freedom of the data sets whereas through p-values, we can obtain the conclusion of whether the null hypothesis can be rejected or supported.
- The p-value of a t-test can be obtained through a manual calculation using the t-distribution tables or through online calculators/ software whereas a p-value can be calculated through an inverse CDF.
Thus, through the above discussions and mathematical expressions, we can conclude that while both these concepts are significantly interlinked, they are not the same and cannot be used interchangeably. Both the concepts provide vital information that can be used for further statistical calculations.