ISS 2016 Statistics Paper-1 Solution: Question 28 (Multiple Correlation Coefficient R1.23)

Continuing our question-by-question walk through the ISS Statistics Paper-1 previous year papers, we now arrive at Question 28 of the 2016 paper. This one sits in the multivariate correlation portion of the syllabus, an area many aspirants find intimidating simply because the notation looks heavier than the actual algebra involved. As always, we will work through every line of the calculation so that nothing is taken on faith.

Quick Summary

• Topic: Multiple correlation coefficient in a trivariate distribution

• Question Reference: ISS 2016, Statistics Paper-1, Question 28

• Correct Answer: Option (c) — √2ρ / √(1+ρ)

The Question, As Asked

In a trivariate distribution, if r12 = r23 = r31 = ρ ≠ 1, then what is the value of R1.23?

• (a) ρ / √(1+ρ)

• (b) 1 / √(1+ρ)

• (c) √2ρ / √(1+ρ)

• (d) 1 / (1+ρ)

Step 1: What R1.23 Actually Means

Before touching any formula, it helps to be precise about what the symbol R1.23 stands for. When we have three variables X1, X2, X3, we can ask: "how well can X1 be predicted using a linear combination of X2 and X3 together?" The answer to that question is a single number between 0 and 1, called the multiple correlation coefficient, written R1.23. It is simply the ordinary correlation between the actual values of X1 and the values predicted for X1 by the best-fitting linear regression on X2 and X3. Unlike a simple correlation coefficient rij, which can be negative, R1.23 is always taken as non-negative, because it measures the strength of a joint linear relationship, not its direction.

Step 2: Writing Down the Correlation Matrix

For three standardized variables X1, X2, X3, the correlation matrix R collects all the pairwise correlations:

• Row 1: 1, r12, r13

• Row 2: r12, 1, r23

• Row 3: r13, r23, 1

The diagonal entries are 1 because any variable is perfectly correlated with itself, and the matrix is symmetric because rij = rji. In this question we are told r12 = r23 = r31 = ρ, and since r31 = r13, all three off-diagonal correlations are equal to the same value ρ.

Step 3: The General Formula for R1.23

There is a standard identity that every student preparing for this exam should be able to derive on demand, not just recall:

R21.23 = 1 − |R| / |R22|

Here |R| is the determinant of the full 3×3 correlation matrix, and |R22| is the determinant of the 2×2 correlation matrix formed only by the predictor variables X2 and X3, which is simply 1 − r232.

Let us compute |R| by expanding along the first row:

|R| = 1·(1 − r232) − r12·(r12 − r23r13) + r13·(r12r23 − r13)

Expanding each bracket term by term:

|R| = 1 − r232 − r122 + r12r23r13 + r13r12r23 − r132

The two middle terms are identical (both equal r12r13r23), so they combine into one term multiplied by 2:

|R| = 1 − r122 − r132 − r232 + 2r12r13r23

Now substitute this into the R1.23 formula:

R21.23 = 1 − [1 − r122 − r132 − r232 + 2r12r13r23] / (1 − r232)

Putting everything over the common denominator (1 − r232) and simplifying the numerator carefully:

R21.23 = [(1 − r232) − 1 + r122 + r132 + r232 − 2r12r13r23] / (1 − r232)

The −1 and the "+1 hidden inside −r232 + r232" cancel, leaving the well-known standard result:

R21.23 = (r122 + r132 − 2r12r13r23) / (1 − r232)

This is the formula worth memorizing, but notice we did not simply pull it out of thin air — we built it from the determinant definition, so even if you forget it under exam pressure, you can reconstruct it in under a minute.

Step 4: Substituting the Given Values

The question tells us r12 = r13 = r23 = ρ (remember r31 = r13). Substituting all three correlations as ρ:

R21.23 = (ρ2 + ρ2 − 2·ρ·ρ·ρ) / (1 − ρ2)

R21.23 = (2ρ2 − 2ρ3) / (1 − ρ2)

Step 5: Simplifying to the Final Answer

Factor 2ρ2 out of the numerator and factor the denominator as a difference of squares:

Numerator: 2ρ2(1 − ρ)

Denominator: (1 − ρ)(1 + ρ)

So:

R21.23 = 2ρ2(1 − ρ) / [(1 − ρ)(1 + ρ)]

Since we are explicitly told ρ ≠ 1, the factor (1 − ρ) is never zero, so it is safe to cancel it from top and bottom:

R21.23 = 2ρ2 / (1 + ρ)

Taking the square root of both sides (recall R1.23 is always taken as non-negative):

R1.23 = √(2ρ2 / (1 + ρ)) = (ρ√2) / √(1 + ρ) =√2ρ / √(1+ρ)

Pro Tip: The single easiest way to lose marks on this type of question is to forget why ρ ≠ 1 was mentioned in the problem statement at all. It is not decoration — it is the exam setter's signal that you are expected to cancel a (1 − ρ) factor between numerator and denominator, and that cancellation is only legal because ρ is guaranteed not to equal 1. The way this derivation is usually taught is to always factor both the numerator and denominator fully before simplifying a ratio like this, rather than trying to simplify term by term, because it makes exactly this kind of cancellation visible instead of easy to miss under time pressure.

This gives us option (c): √2ρ / √(1+ρ), which matches the official ISS 2016 answer key exactly (Question 28 is marked as C).

Why This Question Matters

Multiple correlation questions like this one test three separate skills at once: your comfort with correlation matrices and determinants, your ability to substitute symmetric conditions (all three correlations equal) without making an error, and your algebraic care in factoring and cancelling. It is also a very "recyclable" question type — the ISS and allied statistical exams have asked variations of this exact trivariate symmetric-correlation setup multiple times, sometimes asking for R1.23 and sometimes for the partial correlation r12.3 instead. Students who drill this kind of derivation until the determinant expansion becomes automatic tend to solve such questions in under two minutes in the actual exam, instead of trying to recall a formula sheet from memory.

Frequently Asked Questions

What is the difference between multiple correlation and partial correlation?

Multiple correlation R1.23 measures how well X1 is predicted by X2 and X3 jointly, as a single combined measure. Partial correlation r12.3, on the other hand, measures the correlation between X1 and X2 alone, after removing the linear influence of X3 from both of them. They answer different questions and use different formulas, though both are built from the same correlation matrix.

Why is R1.23 always non-negative even though r12 or r13 could be negative?

R1.23 is defined as the correlation between the actual X1 values and the values predicted by the regression on X2 and X3. A good linear fit will always be positively correlated with the actual data it is fitting, by construction, so by convention the positive square root is always taken when computing R1.23 from R21.23.

Can R1.23 ever exceed 1?

No. Since R1.23 is itself a correlation coefficient, it must lie between 0 and 1 inclusive. If your algebra ever produces a value greater than 1, that is a strong signal to recheck your substitution or your sign in the determinant expansion.

What happens to this formula if rho equals exactly 1?

If ρ = 1, all three variables would be perfectly linearly related to one another, which makes the correlation matrix singular (its determinant becomes zero) and the (1 − ρ) cancellation in Step 5 would be dividing zero by zero. That is precisely why the question specifies ρ ≠ 1 — it rules out this degenerate case.

Is it necessary to memorize the R1.23 determinant formula for the exam?

It helps to recognize it on sight, but it is far more valuable to know how to derive it from the 3×3 correlation matrix determinant, as shown in Step 3 above. Examiners frequently tweak the setup (unequal correlations, four variables instead of three, or asking for partial correlation instead), and a memorized formula alone will not adapt to those variations the way understanding the derivation will.

How is this topic typically weighted in the ISS Statistics Paper-1 exam?

Correlation and regression, including multiple and partial correlation, are core topics that appear almost every year in some form, often as one or two objective questions. Given how mechanical the algebra becomes once the determinant-based formula is internalized, it is one of the more reliably scorable topics if you have practiced it thoroughly.

If any step above felt unclear, drop a comment below and we will work through it together — and if you found this useful, do share it with a fellow ISS aspirant who might be revising the same topic.