ISS 2016 Statistics Paper-1 Solution: Question 36 (Partial Correlation Coefficient Identities)
Continuing our question-by-question walk through the ISS Statistics Paper-1 archive, today we land on a question from the 2016 paper that ties together three separate ideas about partial correlation into a single "which of these are true" puzzle. If you have been following this series in order, you will notice the questions are getting progressively more theory-heavy — this one rewards a reader who can actually derive a formula rather than just recall it.
Quick Summary
Topic: Partial correlation coefficient — its formula, its bound, and its link to partial regression coefficients
Question Reference: ISS 2016, Statistics Paper-1, Question 36
Correct Answer: Option (d) — Statements 1, 2 and 3 are all correct
The Question
If r12.3 is the correlation coefficient between the variables X1 and X2 after eliminating the linear effect of X3, then which of the following are correct?
r12.3 = (r12 − r13r23) / √[(1 − r132)(1 − r232)]
r122 + r232 + r312 − 2r12r23r31 ≤ 1
r12.32 = b12.3b21.3, where b's are partial regression coefficients
Options:
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
Step 1: What Does "Eliminating the Linear Effect of X₃" Actually Mean?
Before checking any statement, we need to be completely clear on what r12.3 is. It is not just r12 recalculated on a smaller dataset — it is the correlation between what is left over in X1 and X2 once you strip away whatever part of each of them can be explained by X3.
To make the algebra clean, assume X1, X2, X3 are standardized — mean 0, variance 1. This costs us nothing because correlation coefficients don't change under such scaling, and it removes a lot of clutter from the working.
Regress X1 on X3. Since both are standardized, the best-fit line is simply X̂1 = r13X3. The part of X1 that has nothing to do with X3 is the residual:
e1 = X1 − r13X3
Similarly, the part of X2 left over after removing X3's influence is:
e2 = X2 − r23X3
By definition, r12.3 is simply the ordinary correlation coefficient between e1 and e2.
Step 2: Deriving the Formula in Statement 1
We need Cov(e1, e2), Var(e1) and Var(e2). Remember, since the variables are standardized: E[Xi] = 0, E[Xi2] = 1, and E[XiXj] = rij.
Covariance of the residuals:
E[e1e2] = E[(X1 − r13X3)(X2 − r23X3)]
Expand this out term by term, just like multiplying two brackets in ordinary algebra:
= E[X1X2] − r23E[X1X3] − r13E[X3X2] + r13r23E[X32]
= r12 − r23r13 − r13r23 + r13r23(1)
= r12 − r13r23
Variance of e1:
Var(e1) = E[(X1 − r13X3)2] = E[X12] − 2r13E[X1X3] + r132E[X32] = 1 − 2r132 + r132 = 1 − r132
By exactly the same steps, Var(e2) = 1 − r232.
Putting it together:
r12.3 = Cov(e1,e2) / √[Var(e1)Var(e2)] = (r12 − r13r23) / √[(1 − r132)(1 − r232)]
This is exactly Statement 1.Statement 1 is correct.
Step 3: Checking the Inequality in Statement 2
This looks like a random algebraic identity at first glance, but it actually falls straight out of what we just derived — there is no need to memorize it separately.
A correlation coefficient of any two real random variables must satisfy −1 ≤ correlation ≤ 1. Since r12.3 is itself a genuine correlation coefficient (of e1 and e2), it must obey the same rule:
r12.32 ≤ 1
Substitute the formula from Step 2:
(r12 − r13r23)2 / [(1 − r132)(1 − r232)] ≤ 1
Since (1 − r132) and (1 − r232) are both positive (they are variances of residuals, and variances can't be negative), we can multiply both sides by that product without flipping the inequality:
(r12 − r13r23)2 ≤ (1 − r132)(1 − r232)
Expand both sides carefully. Left side:
r122 − 2r12r13r23 + r132r232
Right side:
1 − r132 − r232 + r132r232
The r132r232 term appears on both sides, so it cancels out. What remains:
r122 − 2r12r13r23 ≤ 1 − r132 − r232
Move every term to the left side:
r122 + r132 + r232 − 2r12r13r23 ≤ 1
Since r31 is exactly the same number as r13 (correlation is symmetric — the correlation between X3 and X1 is identical to the correlation between X1 and X3), this is precisely Statement 2.Statement 2 is correct.
Pro Tip: This inequality is where most candidates lose marks — not because the algebra is hard, but because they try to treat it as a standalone fact to cram, usually by connecting it to "positive semi-definiteness of the correlation matrix" without seeing where that condition actually comes from. The more reliable route, and the one that's usually drilled into students until it becomes automatic, is to recognize that r12.32 ≤ 1 is the entire inequality in disguise — once you know the partial correlation formula from Statement 1, Statement 2 is just one substitution and one expansion away, with no new theory required. This habit of hunting for "is this secretly the same fact as the previous part?" is a big time-saver in a paper like this one, and it's a pattern this style of teaching — the kind you'll find at a place like Sunrise Classes — leans on constantly.
Step 4: Verifying the Regression-Coefficient Identity in Statement 3
Now we need the partial regression coefficients b12.3 and b21.3. These come from multiple regression, not simple regression, so let's build them carefully.
b12.3 is defined as the coefficient of X2 when X1 is regressed on both X2 and X3 together (i.e., X3's effect is held fixed while X2's effect on X1 is measured). For standardized variables, the general two-predictor regression formula gives:
b12.3 = (r12 − r13r23) / (1 − r232)
Similarly, b21.3 is the coefficient of X1 when X2 is regressed on both X1 and X3:
b21.3 = (r12 − r23r13) / (1 − r132)
Notice that the numerator, (r12 − r13r23), is identical in both — only the denominators differ, because the "third variable being held fixed" swaps role each time. Now multiply the two:
b12.3 × b21.3 = (r12 − r13r23)/(1 − r232) × (r12 − r13r23)/(1 − r132) = (r12 − r13r23)2 / [(1 − r132)(1 − r232)]
Compare this to what we already derived in Step 2 for r12.32 — it's the exact same expression. So:
r12.32 = b12.3b21.3
This is precisely Statement 3, and it is the natural three-variable extension of the familiar two-variable fact that r2 = byxbxy (the product of the two simple regression coefficients equals the square of the correlation coefficient).Statement 3 is correct.
Step 5: Final Answer
All three statements — the formula, the bound, and the regression-coefficient identity — check out under direct derivation. So the correct option is:
(d) 1, 2 and 3
This matches the official ISS 2016 Statistics Paper-1 answer key, which lists the answer to Question 36 as D. Our independent derivation and the official key agree completely.
Why This Question Matters
Partial correlation sits at the junction of correlation theory and multiple regression, and ISS papers return to it again and again in different disguises — sometimes as a direct formula question like this one, sometimes buried inside a numerical problem asking you to compute r12.3 from given simple correlations. What makes this particular question valuable practice is that it forces you to hold three related identities in your head simultaneously and see how one implies the next, rather than memorizing each in isolation. Once you've derived all three from scratch even once, you stop needing to memorize any of them — they simply become "the same algebra, viewed from three angles."
Frequently Asked Questions
What exactly is the difference between a simple correlation and a partial correlation?
Simple correlation r12 measures the raw linear association between X1 and X2, including any association that exists only because both happen to be related to a third variable X3. Partial correlation r12.3 strips out X3's influence first, by working with the residuals of X1 and X2 after regressing each on X3, so it reflects the association between X1 and X2 that remains once X3 is "controlled for."
Why do we bother "eliminating" the effect of a third variable at all?
Two variables can appear strongly correlated purely because they are both driven by a common third factor — a classic example being how both ice cream sales and drowning incidents rise in summer, correlated with each other only because both are correlated with temperature. Partial correlation lets a statistician check whether X1 and X2 are genuinely related once that shared driver is accounted for.
Can r₁₂.₃ ever be larger than 1 in magnitude?
No. Since r12.3 is defined as an ordinary correlation coefficient between two residual variables (e1 and e2), it must obey the same universal rule as any correlation coefficient: its value always lies between −1 and 1. That single fact is exactly what let us derive Statement 2 in this question without any extra memorization.
What is the difference between a partial correlation coefficient and a partial regression coefficient?
The partial correlation coefficient r12.3 is symmetric — it treats X1 and X2 identically and is always between −1 and 1. A partial regression coefficient like b12.3 is not symmetric — it depends on which variable is treated as the outcome and which as the predictor, and it also carries scale information (it can be any real number), which is why Statement 3 needs the product of two different partial regression coefficients to recover the square of the symmetric partial correlation.
Does the inequality in Statement 2 have a deeper meaning beyond just being "true"?
Yes — it is equivalent to saying that the 3×3 correlation matrix of X1, X2, X3 has a non-negative determinant, which is a basic requirement for any valid correlation matrix (it must be positive semi-definite). Any three correlation values that violate this inequality could never actually arise from real data, no matter how large the sample.
Is the formula for r₁₂.₃ specific to exactly three variables?
The specific formula used here is the first-order partial correlation, involving exactly one variable being controlled for. Higher-order partial correlations, which control for two or more variables simultaneously, extend the same residual-based idea but use a more general recursive formula built on lower-order partial correlations.
That covers Question 36 of the ISS 2016 Statistics Paper-1. If any step in the derivation felt rushed or you'd like it worked through differently, drop a comment below — and if you know a fellow ISS aspirant working through the same paper, consider sharing this with them before the next question in the series goes up.
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