Question
Suppose $r_{1.23}$ and $r_{1.234}$ are sample multiple correlation coefficients of $X_1$ on $X_2, X_3$ and $X_1$ on $X_2, X_3, X_4$ respectively. Which of the following is possible?
The correct answer is
$r_{1.23} = 0.3, r_{1.234} = 0.7$
Multiple Correlation Coefficient Properties
The sample multiple correlation coefficient, denoted here as $r$, measures the strength of the linear relationship between a dependent variable ($X_1$) and a set of predictor variables (e.g., $\{X_2, X_3\}$ or $\{X_2, X_3, X_4\}$).
Key properties of the multiple correlation coefficient ($R$) include:
- Non-negativity: The multiple correlation coefficient is always non-negative and cannot exceed 1. Mathematically, $0 \le r \le 1$.
- Monotonicity: Adding more predictor variables to the model can only increase or maintain the multiple correlation coefficient; it cannot decrease it. Therefore, when comparing $r_{1.23}$ (correlation with $X_2, X_3$) and $r_{1.234}$ (correlation with $X_2, X_3, X_4$), it must be that $r_{1.234} \ge r_{1.23}$.
Analyzing Coefficient Possibilities
Let's evaluate the given options based on these two properties:
- Option 1: $r_{1.23} = -0.3, r_{1.234} = 0.7$. This is impossible because multiple correlation coefficients cannot be negative (violates the non-negativity property).
- Option 2: $r_{1.23} = 0.7, r_{1.234} = 0.3$. This is impossible because adding variable $X_4$ decreased the correlation coefficient ($0.3 < 0.7$), violating the monotonicity property.
- Option 3: $r_{1.23} = 0.3, r_{1.234} = 0.7$. This is possible. Both values are within the range $[0, 1]$, and the coefficient increased ($0.7 \ge 0.3$) when an additional variable ($X_4$) was included.
- Option 4: $r_{1.23} = 0.7, r_{1.234} = -0.3$. This is impossible because the multiple correlation coefficient cannot be negative (violates the non-negativity property).
Conclusion
Based on the properties that multiple correlation coefficients must be between 0 and 1, and that they cannot decrease when more variables are added, only Option 3 presents a statistically possible scenario.
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Important Questions from Ratio And Regression
- A simple random sample (without replacement) of size $n$ is drawn from a finite population of size $N (\ge 7)$. What is the probability that the $4^{\text{th}}$ population unit is included in the sample but the $6^{\text{th}}$ population unit is not included in the sample?
- For a data set $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ the following two models were fitted using least square method.
Model 1: $y_i = \beta_0 + \beta_1 x_i \quad i = 1, 2, \dots n$
Model 2: $y_i = \beta_0^* + \beta_1^* x_i + \beta_2^* x_i^2 \quad i = 1, 2, \dots n$
Let $\hat{\beta}_0, \hat{\beta}_1$ be least square estimates of $\beta_0, \beta_1$ from model 1 and $\hat{\beta}_0^*, \hat{\beta}_1^*, \hat{\beta}_2^*$ be the least square estimates from model 2.
Let $A = \sum_1^n \left(y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_i)\right)^2$,
$B = \sum_1^n \left(y_i - (\hat{\beta}_0^* + \hat{\beta}_1^* x_i + \hat{\beta}_2^* x_i^2)\right)^2$
Then - Suppose $\bar{Y}$ is the sample mean of the study variables corresponding to a sample of size n using simple random sampling with replacement scheme and $\bar{Y}_{st}$ is the sample mean of the study variables corresponding to a sample of size n using stratified random sampling with replacement scheme under proportional allocation. Which of the following is/are sufficient condition/conditions for $Var(\bar{Y}) = Var(\bar{Y}_{st})$?
- Consider the problem of drawing a sample of size 2 from a finite population of size 20. The sampling is done with replacement using probability proportional to size sampling scheme. The normed size measures $p_1, \cdots, p_{20}$ are given by $p_i = \frac{1}{40}$, $i = 1, \cdots, 10, \; p_i = \frac{3}{40}$, $i = 11, \cdots, 20$. The expected number of distinct units drawn is
- Consider a finite population of size $N$. Let $T_1$ be the sample mean based on a sample of size $n$ under simple random sampling with replacement (SRSWR) scheme. Let $T_2$ be the sample mean based on a stratified random sample of size $n$ where the samples are drawn from each of 4 strata using SRSWR scheme under proportional allocation. Then which of the following are sufficient conditions for $\text{Var}(T_1) = \text{Var}(T_2)$ to hold?
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