In List I types of variables are given and in List II correlation coefficients related to them are given. Match List I and List II and choose the correct answer from the code given below. List I (Types of variable) List II (Correlation coefficient)a) Both variables continuous i) Biserial b) Both with ordinal measure ii) Pearson's 'r' c) Both with artificial dichotomy iii) Contingency coefficient d) One continuous and other with artificial dichotomy iv) Spearman's rho e) Two or more categories v) Tetrachoric 'r' vi) Point-Biserial
a-ii, b-iv, c-v, d-i, e-iii
Understanding Correlation Coefficients and Variable Types
Correlation coefficients are powerful tools in statistics used to measure the strength and direction of the relationship between two variables. The type of correlation coefficient you should use depends heavily on the scale of measurement of the variables you are analyzing.
Let's break down the relationship between the types of variables listed in List I and the appropriate correlation coefficients from List II:
- a) Both variables continuous: When both variables are measured on a continuous scale (like interval or ratio scales), the most appropriate coefficient is Pearson's 'r' ($\rho_{xy}$). This coefficient measures the strength and direction of a linear relationship between the two variables. This corresponds to option (ii) in List II.
- b) Both with ordinal measure: If both variables are measured on an ordinal scale (meaning they represent ranks or ordered categories but the distance between values isn't uniform), Spearman's rho ($\rho$) is the suitable choice. It assesses the strength and direction of a monotonic relationship between the variables. This matches with option (iv) in List II.
- c) Both with artificial dichotomy: An artificial dichotomy is a variable that is inherently continuous but has been categorized into two groups (e.g., 'high' vs. 'low' performance based on a test score). When both variables being correlated are artificial dichotomies, the Tetrachoric 'r' is used. It estimates what the Pearson correlation would be if the underlying continuous variables were available and normally distributed. This aligns with option (v) in List II.
- d) One continuous and other with artificial dichotomy: When you have one variable measured on a continuous scale and the other is an artificial dichotomy, the Biserial correlation (r$_{bis}$) is the correct coefficient. It also assumes that the dichotomous variable represents an underlying continuous variable with a normal distribution. This matches with option (i) in List II.
- e) Two or more categories: For assessing the association between two categorical variables, especially when they have two or more categories (nominal scale), the Contingency coefficient (C) is used. It is derived from the chi-square statistic and indicates the degree of association. This corresponds to option (iii) in List II.
Matching Summary Table
| List I (Types of Variable) | List II (Correlation Coefficient) | Match |
|---|---|---|
| a) Both variables continuous | ii) Pearson's 'r' | a-ii |
| b) Both with ordinal measure | iv) Spearman's rho | b-iv |
| c) Both with artificial dichotomy | v) Tetrachoric 'r' | c-v |
| d) One continuous and other with artificial dichotomy | i) Biserial | d-i |
| e) Two or more categories | iii) Contingency coefficient | e-iii |
Based on the analysis of variable types and their corresponding correlation coefficients, the correct matching is a-ii, b-iv, c-v, d-i, e-iii.
Revision Table: Correlation Coefficients Explained
| Coefficient | Variable Types | Key Use Case |
|---|---|---|
| Pearson's r | Continuous vs. Continuous | Linear relationship |
| Spearman's rho | Ordinal vs. Ordinal (Ranks) | Monotonic relationship, non-parametric |
| Tetrachoric r | Artificial Dichotomy vs. Artificial Dichotomy | Estimates Pearson's r assuming underlying continuous normal distribution |
| Biserial | Continuous vs. Artificial Dichotomy | Estimates Pearson's r assuming underlying continuous normal distribution for the dichotomy |
| Point-Biserial | Continuous vs. True Dichotomy | Used for naturally occurring dichotomies (e.g., male/female, pass/fail) |
| Contingency Coefficient | Categorical (two or more categories) vs. Categorical | Measures association between nominal variables |
Additional Information on Dichotomous Variables
It's important to distinguish between true and artificial dichotomies when choosing a correlation coefficient involving a continuous variable:
- True Dichotomy: A variable that is inherently two-valued and cannot be seen as representing an underlying continuous dimension (e.g., gender, alive/dead, yes/no). The Point-Biserial correlation is used between a continuous variable and a true dichotomy.
- Artificial Dichotomy: A variable that is continuous in nature but has been split into two categories based on some criterion (e.g., scoring above or below a median on a test, classifying height as 'tall' or 'short'). The Biserial correlation is used between a continuous variable and an artificial dichotomy.
Understanding this distinction helps in applying the correct statistical test and interpreting the results accurately.
Important Questions from Measurement and Analysis of Data - Teaching
While wandering in jungle, King Dushyant married Shankuntala as narrated in the "Abhigyan Shankuntalam". He gave her royal ring which could serve her identity when she would come to meet him, in future. However, she had lost the ring while going to meet him. When she arrived at Dushyant's palace, he failed to recognise her as she did not had the ring.
Which of the following statistical error King Dushyant had committed in this narrative?
An investigator used ANOVA to compare four groups of students on numerical ability on the basis of a test. After analysis of raw scores, the following results were obtained:
Source of variation df Sum of Squares Between Groups 3 625.00 Within Groups 36 2128.00 The value of F-ratio would be approximate:
In randomly constituted two groups-experimental and control, a researcher obtains the following results after using a parametric 't' test:
Value of t = 3 for N = 300
On the basis of this evidence which decision in respect of substantive research hypothesis and the null hypothesis will be justified?
Given below are two statements, one labelled as Assertion (A) and the other labelled as Reason (R). Read the statements and choose the correct answer using the code given below.
Assertion (A): Homogenous tests have low reliability.
Reason (R): The range of test scores affects reliability.
An investigator commits Type I error in testing hypothesis when he / she
