It is not surprising, it is sufficnet that one (or more) of the covariate is (even slightly) negatively correlated with Y and positively with X (or viceversa, in any case pointing to the opposite direction with respect to the main correlation flux) that partialling it out the correlation increases.

I think you should find a good statistics text and read about what it means to "control for other covariates" in a statistical model. Although you're interested specifically in the "partial correlation", you may find it easier to being with the  regression context and the meaning of 'partial regression coefficients" = expected change in y for a unit change in the x of interest WHEN ALL OTHER COVARIATES ARE HELD CONSTANT.  Think about what this means in terms of eliminating sources of variability.

At an introductory level: Martin Bland's Introduction to Basic Statistics or Kirkwood's Essential Biostatistics both explain this. I know this is not the first time I've suggested consulting a basic stats text, so I hope you will follow through. Because there are some things that are best understood by thinking about them alone in a quiet room :)

-----------------------------------------------

Atul Sharma MD, MSc, FRCPC, M.Stat

Senior Consultant | Biostatistics Group | Data Science Platform

George & Fay Yee Centre for Healthcare Innovation

In addition to my previous note, you should also realize that your "statistical significances" is a test of the correlation against the null hypothesis of 0. That's not the same as saying the increase from 0.79 to 0.92 was statistically significant, though this can be easily checked using the Fisher Z transform or likelihood ratio test. 

@Alessandro-- Thank for explanation. That means, those controlled variables exerted a negative correlation before I partial them out. That why I m getting a positive correlation after controlled those covariates. Is that valid to say "This indicates that all 3 covariates had very little influence in controlling for the relationship between exhaust-venous isoflurane. "

@Atul- Hi Atul. I am using the textbook (Introduction to SPSS statistics in Psychology Version19 and earlier - 5th edition by Dennis Howitt and Duncan Cramer) and online resource *https://statistics.laerd.com/spss-tutorials/partial-correlation-using-spss-statistics.php*

I learnt the reporting methods via the Laerd website. 

Yes, more or less, the importance is in the sign...but please go ahead from simply consulting a program manual for learning statistics, try to understand the meaning of what you are doing by both going more in depth in your readings and by stop and thinking of what you are doing. Stp and thinking so to aquire a correct link between the methods and your representation of reality is by far more important of both studying and doing analyses by pushing the right button on a keyboard. This is of paramount importance in statistics that can be roughly described as 'good common sense + a little bit of mathematics'.

I'm not sure you're paying sufficient attention to the details of the test.  The default null hypothesis is simply r = 0. So 0.72 !=0 (p=0.011) and 0.92 !=0 (p=0.008). But it doesn't follow that these two numbers are different from each other, since you haven't tested the equality of the two correlation coefficients, which may or may not be different depending on the sample size. In which case, all this talk of "improving with adjustment" may be spurious (chance associations in your sample). The comparison of two r values  is so simple that I often do it by hand (assuming normality after Fisher Z transform), which requires only the r values and respective sample sizes. Since I don't know your sample size, I can't comment except to say that those two numbers are probably not statistically different unless you have more than 40 measurements. A likelihood ratio test for nested models would be a more formal test for differences. 

I'm not trying to be a curmudgeon, but inappropriate applications of statistical tests is dangerous: 

In April, the editor of the Lancet (Richard Horton) wrote:

…much of the scientific literature, perhaps half, may simply be untrue. Afflicted by studies with small sample sizes, tiny effects, invalid exploratory analyses, and flagrant conflicts of interest, together with an obsession for pursuing fashionable trends of dubious importance, science has taken a turn towards darkness. 

@Atul- I got my information from this website (https://statistics.laerd.com/spss-tutorials/partial-correlation-using-spss-statistics.php)

Reporting Guide:

"A partial correlation was run to determine the relationship between an individual's VO2max and weight whilst controlling for age. There was a moderate, negative partial correlation between VO2max (43.63 ± 8.57 ml/min/kg) and weight (79.66 ± 15.09 kg) whilst controlling for age (31.1 ± 9.1 years), which was statistically significant, r(97) = -.314, N = 100, p = .002. However, zero-order correlations showed that there was a statistically significant, moderate, negative correlation between VO2max and weight (r(98) = -.307, n = 100, p < .002), indicating that age had very little influence in controlling for the relationship between VO2max and weight."

My sample sizes is 9. But I cant help with this as my study duration is only limited to 3months. 

When you control for covriates, the relationship between two variables may increase or decrease.   For example, suppose we have three variables say Intelligent Quotient (IQ), Mental Age (MA) and Anxiety (A). Suppose ordinary correlation between IQ and MA is 0.603. Now you control for Anxiety (A) and again compute correlation between IQ and MA and let it's value is 0.92. It means that the correlation between IQ & MA Can improve significantly if Doctor's able to control the Anxiety of the patients.

Dear Suresh, 

It is such a clear cut example. Thank you so much. 

What about to explain the sig value? 

"There were statistically significant in the correlation of exhaust-arterial isoflurane, r (7)= .874, p= .002 . In the analysis of partial correlation by controlling covariates (namely:- haemoglobin, mixed venous temperature and gas flow rate), the coefficient increased to r(4)= .998, p=.001 in exhaust-arterial isoflurane . "

That means if i can control covariates above, the correlation will be getting more positively correlated. How to explain that p=.001? How I know whether are those 3 covariates signifacantly affect my correlation exhuast-artery?

Many thanks.

Given the sample size I don't think you have sufficient statistical power to detect an increase in r following patrolling out covariates. For instance the 95% CI for your correlation of .874 is [.44, .98]. At best you could argue that the correlation remains high after adjusting for them.

Hi Thom, 

I explained it as below:

"The partial correlation analysis in our study has established that the correlation between blood isoflurane and exhaust isoflurane remained statistically significant after partialling out the influences of the above covariates at the point where the blood samples were taken. Although the sample size was relatively small, there did not appear to have obvious outliers in the regression lines of exhaust isoflurane and its blood levels."

Would this make any sense to you given my sample size is small and the weakness of partial correlation test as you mentioned?

Many thanks.

KT

Yes - that seems reasonable (though I'm not an expert on these topics - sounds like anaesthetics). I would actually plot the relationship myself ... The partial correlation test isn't weak per se but you lose df and hence statistical power quite quickly when you start with only 8 pairs.

A more accurate though less used measure of multiple correlation that makes more sense in multiple regression is the semi-partial correlation. Which is like the partial correlation except that at the denominator we do not divide by the second sqrt(1-r^2(X,Z) ). For the case of X,Y,Z see Huber, J., 1981. Partial and semipartial correlation—A vector approach. Two-Year College Math. J. 12, 151–153.

The extension of the semi-partial correlation to n dimension yields the elements of Cholesky factor of a correlation matrix.

Consider the 3 dim (X,Y,Z) case: It is enough that sqrt(1-r^2(X,Z) ) is so small, or that |r(X,Z)| is very near 1 to make the partial correlation very large.

On the other hand, it could be that |r(X,Y)| is very big or that |r(Y,Z)| is larger than both |r(X,Y)| and |r(X,Z)}

.

With only 8 pairs, I suspect the difference between 0.79 and 0.92 isn't statistically significant. You would need to formally compare the two models using some form of likelihood ratio test , but a quick look using the Fisher Z transform tells me that these two numbers can't be distinguished statistically with fewer than 20 pairs  It's important to realize that your p-values refer, in each case, to the null hypothesis of 0 i.e. 0.72 0 and 0.92  0, but you haven't compared 0.72 to 0.92 yet. You have to be careful with p-values, to remember what the null hypothesis they're testing.

Atul, I agree that with such a small sample size, the observed difference is unlikely to achieve statistical significance.  But I am curious about how you are testing the difference between a zero-order correlation, and a partial correlation computed using the same cases.  Obviously, the two correlations are not independent of each other, and the dependency will have to be taken into account when computing the SE of the difference.  Can you provide more details, or a reference?  Thanks. 

@Vered- Thank you for your comment. What is the indication to use partial or semi partial correlation? Is there any assumption to be met?

@Atul- Thank you for your feedback. All the variables were measured in the same patients (exhuast isoflurane, venous isoflurane,haemoglobin, mixed venous temperature and gas flow rate)-- all variables were normally distributed wiht Shapiro Silk test.Apologies for my low quality question, which is I did not see the indication to use Fisher Z transform. it would be valuable to gain some explanation from you.

@Bruce- Many thanks for your input. I am not getting what you meant by same case? 

Exhaust isoflurane -- venous isoflurane (r= 0.79, p= 0.011)

Exhaust isoflurane -- venous isoflurane after controlling haemoglobin, mixed venous temperature and gas flow rate (r= 0.92, p= 0.008)

@Ka Ting Ng:  I said the zero-order and partial correlations were computed using the same cases, meaning data from the same subjects.  Therefore, the two correlations are related, not independent.  HTH.

Sorry, I was most definitely not suggesting the Fisher test was suitable for correlated outcomes. But as I noted, you can certainly do a likelihood ratio test (same data, nested models, partialling out the other covariates) with a random effects to account for repeated measures, as detailed in any textbook on mixed effects models. I was merely noting that the back-of-the envelope calculation suggests that it is going to be difficult to distinguish these small differences. Yes, if the results are positively correlated (and I assume they are), the required N will be smaller. But I can't pre-judge the test, and with a sample this limited, I don't think we can safely assume that it's a meaningful difference. 

Ka Tin Ng,

No special assumptions than what you have when you use the partial correlation.

Let's write down your linear model:

Y = Exhaust isoflurane

X1 = haemoglobin,

X2 = mixed venous temperature

X3 = gas flow rate

X4=  venous isoflurane ( r(Y,X4) = 0.79, p= 0.011)

Your partial correlation is r = r_pc(Y,X4  | X1, X2,X3) = 0.92 did you use Fisher Z transform to compute the p-value? Or you used a t test to test the partial correlation? 

T = r/sqrt(1-r^2) *sqrt(N-5) which is t with N-5=df if rho_pc(Y,X4 | X1, X2,X3) = 0.

So your T = .92/sqrt(1-.92^2)*sqrt(N-5) N is the total no of samples.

you can do the same for the semi partial correlation

The semi-partial correlation will be  

r_spc(Y,X4 | X1,X2,X3)  = r_pc(Y,X4 | X1, X2,X3) * sqrt(1 - r_spc^2(Y,Y|X1,X2,X3))

So the value of r_spc must be smaller than ,92

Can I get from you the 5x5 correlation matrix of X1,X2,X3,X4,X5=Y ? I can explain better than.

The component [5,4] in  the Cholesky factor  will be that semi-partial correlation you want. Look at  "Direct formulation to Cholesky decomposition of a general

nonsingular correlation matrix" in Stat and Prob letters, 2015.

hello man,

My name is Wenjie Zhang, supervised by Professor Huete Alfredo in UTS. Now I am Ph.D. student, focusing on the interaction between vegetation and climate change.

Considering there is a strong relationship between climate variables, we often use the partial correlation to try to remove that correlation.

I used to explore the relationship between daytime-temperature-vegetation and nighttime-temperature-vegetation based on the partial correlation, controlling the solar radiation and precipitation simultaneously. As all we know, there is a huge correlation especially between daytime and nighttime, which even could reach nearly 1 (figure 1).

However, I encountered a weird problem beyond my knowledge. In my case, I used two different ways to calculate the partial correlation between vegetation and climate variables.

The first one is all variables participate the calculation including day-temperature, night-temperature, precipitation and solar radiation. Then we got the partial correlation coefficient

respectively (figure 2). The second is that we conducted reduced partial correlation analyses in which the dependent variables were correlated twice: once against day-temperature, precipitation and solar radiation, and once against night-temperature, precipitation and solar radiation (figure 3). The weird is the direction of the night coefficients of these two methods is almost opposite: positive occupied the main area in the result of the first approach; the second method is reverse. At the same time, we could find they have a nearly similar coefficient direction of both day-temperature and night-temperature of the second method. What's more, the direction of the night-temperature of the second method is same as the simple Pearson correlation.

Now I am in trouble with these two completely opposite direction. which one is the true answer?

So I wonder whether there is some pre-condition or not. For example, two strong correlated variables (r is more than 0.9) can't input into the partial correlation at the same time.

Look forward to your reply. Thanks very much.

Yours Sincerely,

Wenjie Zhang