[Focus 10a] Pearson's Product Moment Correlation Coefficient (PPMCC)

We have seen that a correlation test can be performed using the Spearman's Rank test if one or both
or the variables falls on the Ordinal scale.

Following on from that, if both variables reach the Interval or Ratio scale... and are Normally
distributed we can use the more robust Pearson's test. The actual theory behind the test is different
but the outcome is similar in that we eventually calculate rp (the Pearson correlation) and the possible range is still -1 through zero to +1. Remember, as the rp value approaches +1 or -1 so the correlation gets stronger and at zero there is no association between the variables at all or so it would seem.

General rule: Always bear in mind though that your research designs (or field observations) are rarely so pure that you can state with certainty that the variables you have chosen to measure have not been affected by some other non-recorded influence. Neither animals, people, the weather or business systems are ever that cooperative !! Always be clear but cautious when drawing your conclusions.

One very important difference between the Spearman's test and the Pearson's test is that we have to use the actual values (and cannot resort to using ranks) e.g. £ vs time, weight vs length or hours vs numbers of people or expenditure vs income etc. It is here that we can see that the PPMCC test is going to be more robust as there is no 'condensing' of the data and no 'blending' of arbitrary 'scores' to produce ranking.

Remember to construct a scatter plot before commencing the analysis because it will give you a good early indication of what rp is going to be and will reassure you that you have done the
subsequent calculation correctly.

The manual procedure requires a rather lengthy table to be constructed because, besides values for x and y; we need to calculate x squared, y squared, xy and sigma values for each of these 5 columns as well as sigma x squared and sigma y squared!!

The null hypothesis is tested by comparing our PPMCC test statistic (rp) with a critical value from the tables. As with the Spearman's test, we must accept the alternative hypothesis if the calculated test statistic (rp) falls outside of that +/- range indicated by the critical value.

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The Coefficient of Determination

Because the correlation coefficient is an index, it only gives us an indication or general interpretation of the correlation. Usually this is sufficient but we can also calculate a derivitive value that is much more precise... The Coefficient of Determination is a measure derived from the Pearson correlation coefficient and is in fact, simply the squred value of it. This offers an easy way of getting a clearer idea idea of the true degree of correlation between two variables. It is the percentage of variance in one variable that is accounted for by the variance in the other.

In the chart below there are three situations outlined. In the first there is no correlation so rp and rp squared are both zero. In the second situation, there is an overlap, the correlation is 0.3 and therefore coefficient of determination is 0.3*0.3 = 0.09. This indicates that they share about 9% of the variance betwen themselves.

In the last example, they overlap heavily , the correlation is high and they share about 81% of the variance between themselves.

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Correlation and Significance

Just as with rs, rp indicates the strength or degree of the supposed association between the two variables. The value for the probability (P) relates to H0 and H1. It gives us the ability to judge the likelihood of H1 or H0 representing the true state of affairs.

It is part of the task to try to explain the variation present in a dataset. Thus, the higher the correlation co-efficient value; the more of the variation that becomes 'explained' and of course, the less variation there actually is. In the chart below, we see a high correlation value and a low level of variation.

How sure are we that any measured supposed association is the true state of affairs?

Task: Write out a simple H0 and H1 for the charts below, take note of the Pearson correlation value....
and the significance levels:

Q. Which of your two hypotheses did you accept and why?

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Task: Compare the two rp values with the patterns displayed on the respective charts.

In both cases the direction and slope of the scatter points seem broadly similar. In the first chart, rp = 0.984; an almost perfect positive correlation and the correlation is highly significant (P< 0.01). As the data becomes more 'scattered' (i.e. more variation), even though there is still a clear similarity to the direction and pattern of the data, we become less sure of the significance of the relationship between x and y.

..... It is usual to see that as the correlation diminishes; so does the significance.

 

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Cautionary note 1: There is always the possibility that there is a strong relationship between two variables but that it is not linear. If, for example, the scattergraph produces a definite curve, the low correlation coefficient might be misinterpreted as indicating no strong correlation. However, there may still be a strong correlation but it is not linear. If this occurs, and you suspect that there is a non-linear relationship, you may wish to consider transforming the data first. (See also Focus 11)

 

Cautionary note 2, about 'cause and effect': You must be satisfied that both variables are normally distributed before using PPMCC and the status of the two variables you are using. In the next example, we have placed 'raw materials costs' on the X axis as the independent variable and 'retail price' on the Y axis as the dependent variable. The problem is that we are making certain assumptions about a relationship existing even before we begin the analysis and this is dangerous.

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The manual calculation of PPMCC

At the Britmax motorcycle factory... The retail price of their best-selling bike, the GT 80, has been compared to the raw materials unit costs every 6 months over the past 6 years.....

Raw materials cost (£) [x]	Retail Price (£) [y]
540	975
580	990
580	1040
590	1020
620	1095
640	1045
650	1060
712	1140
740	1100
760	1195
790	1195
805	1225
Totals:..8007	13080
Totals squared: 64,112,049	171,086,400

Note how large some of the derived figures are going to be.....

First plot the graph.... any deductions to be made at this stage? Clue: has the raw material cost altered (as a percentage) when compared to the retail price? In order to maintain their original margin, the latest retail price should be £1454, why do you think it is not?

Q. What can you say about the relationship between values for 'x' and 'y' so far?

The graph suggests that there is a strong positive relationship here between the two variables and there appears to be a specified uni-directional difference. Hence we will adopt a one-tailed approach to the test. If we merely wanted to know if there was any relationship, positve or negative, then a two-tailed approach would be more appropriate.

A note about one and two-tailed tests; This is a system for testing a null hypothesis against different alternative hypotheses. There are three possible situations for the alternative hypothesis 1) the value of the parameter is different from that in the null 2)  "       "      "   "           "       "   greater than     "   "    "     "      " 3)  "       "      "   "           "       "   less than           "   "    "     "      " Case 1) would require a two-taled approach to testing whilst 2) and 3) would a one-tailed approach

We will need to know the number of degrees of freedom when we check the critical values for rp. The d.f. is calculated as the number of data pairs minus 2, i.e. n = 12 here, therefore d.f = 10)

Question: "Is there any correlation between unit cost of raw materials and the retail price? We are not asking "Is one factor dependent upon the other", we are asking "is there a relationship, if there is; is it positive or negative and can we quantify the strength of any relationship?

The null hypothesis will be that there is no correlation between these two variables.

That is.........Ho: rp = 0

The alternative hypothesis will state that there is a correlation between the two variables.

We now need to calculate all the derivatives. In order that the very large figures involved, do not lead to calculator errors, it is legitimate to move the decimal place. In this case, we have moved it two places to the left e.g 540 become 5.4 and 975 become 9.75 etc ...

Raw materials cost (£) [x]	Retail Price (£) [y]
540	975
580	990
580	1040
590	1020
620	1095
640	1045
650	1060
712	1140
740	1100
760	1195
790	1195
805	1225
Totals:..8007	13080
Totals squared: 64,112,049	171,086,400

(The decimal placed has been moved prior to deriving these values)
x squared	y squared	xy
29.16	95.06	52.65
33.64	98.01	57.42
33.64	108.16	60.32
34.81	104.04	60.18
38.44	119.90	67.89
40.96	109.20	66.88
42.25	112.36	68.90
50.69	129.96	81.17
54.76	121.00	81.40
57.76	142.80	90.82
62.41	142.80	94.41
64.80	150.06	98.61
543.32	1433.35	880.65

Do be very clear that you appreciate the difference between ...in the above example, the former would be 5,433,269 and the latter would be 8007 squared = 64,112,049. The former means " the sum of all the x squareds"; the latter means " the sum of all the x's and then square that sum"

The formula to use looks daunting (see below) but once you have completed the table (as above) it is really quite straight forward unless the figures become too large and unmanageable. As with many statistical formulae; it is a question of substitution. This is a clear example where the computer has the advantage over a manual system but it is still important that you appreciate how and why a particular result is obtained before relying too heavily upon an SPSS output.

Here is the full maths workings to show that they are not that complex if you simplify in set stages.
So in total we need 8 mathematical values to do this calculation and it is a good idea to re-tabulate the figures you need before inserting them into the equation....

Sigma xy	880.65	Sigma (y squared)	1433.35
Sigma x	80.07	(Sigma x).... squared	6411.2
Sigma y	130.80	(Sigma y).... squared	17108.6
Sigma (x squared)	543.32	n	12

Thus......

So rp = 0.948; a strong positive correlation. Note that with such large numbers, rounding effects can be quite noticeable. In this instance values for rp could vary from .947 to .951. It is usual to only give a correlation coefficient to three decimal places.

Check back to the scatter plot. Does this result seem to agree with the display?

A strong positive correlation but is the result significant? After all, our sample size is quite small and as we have often warned, do not be tempted to draw definitive conclusions where small sample sizes are concerned unless you have properly tested for significance.

Remember what the confidence (P-value) is telling us in these cases.

We are only ever able to look at samples but it is really the whole population that we are interested
in so we are saying in effect "how confident can we be that our result would accurately reflect the
true situation to be found in the whole population?".

From tables (see below) giving the critical values for rp:
rp critical value [with P(0.05), d.f = 10)] = ± 0.576 (from Wheater and Cook(2000), p220) and displayed fully below:

0.948 > 0.576. In other words, our test statistic is well outside of the critical value and so we must accept the alternative hypothesis viz: there is a significant correlation here. Now read off the P(0.01) critical value and again compare to our test statistic.....

In conclusion we can say that there is a strong positive correlation between the raw materials cost and the retail price and that the result is significant at the 95% level and also at the 99% level.

In simple terms, the higher the costs, the higher the retail price (Care! there may be other factors that
can influence the retail cost such as wages and competition so as always; be very careful about drawing extended conclusions from your research.)

When there are more than two variables being considered, SPSS can give a correlation matrix as part of the output, The results above the principal diagonal (blue below) are duplicated below it (indicated in black below). We deal more fully with the issue of multi-variable analysis in Focus 14 and Focus 15 where the importance of the correlation matrix is highligted. At this stage it is simply necessary to make the link between simple two-variable correlation and multiple correlation.

In the correlation matrix below, there are 6 variables under test so there are 15 pairs of comparisons possible and so there are going to be 15 rp values created:

Variable	A	B	C	D	E	F
A		rp1	rp2	rp3	rp4	rp5
B			rp6	rp7	rp8	rp9
C				rp10	rp11	rp12
D					rp13	rp14
E						rp15
F

In the next example, we have only 3 variables (all clearly parametric ) to consider and so there will be 3 rp values produced....

Fli-Lo plc have planes up to 18 years old in their fleet. They wish to investigate the degree of association between age (in months), annual £ maintenance cost(£'000) and miles flown between overhauls (m 0'000). 30 planes are checked.

You will now have to calculate three different Pearson correlations. You will begin to appreciate at this point, that once we move from dealing with two variables, to dealing with three or more; statistical procedures get more difficult.

Get into the habit of controlling zero's without misplacing decimal points. When seen in a large table; a series of noughts and commas will only confuse. Once you have moved a decimal point for a given variable, do not then change it.

So the first line shown below indicates: Aeroplane 1) 203 months old, maintenance cost: £61,000 p.a. and 700,000 miles flown

Use: SPsmex16 Age of Planes for the analysis

SPSS will generate a 'correlation matrix' if all 3 variables are carried over (in 'Bivariate data')....do not confuse this procedure with 'Multiple correlation'. In other words, the three correlations examined individually will be: X~Y, X~Z & Y~Z.

Here is the first comparison....

Task: Write H0 and H1. Now work out how to plot all three comparisons on one chart as illustrated below.
Do not try to fit any lines!

 

To produce a 3-D plot with SPSS: >>>> Select 'Graphs' from the drop down menu. Select 'Scatter', '3-D' and in the dialogue box; transfer 'CostPA' to the Y axis, 'Monthold' to the X axis and 'Milestrav' to the Z axis. Click 'OK'

Task: Comment upon the three rs values you have obtained. Note the critical values....
From tables: P (0.05)(28d.f = 0.361) & P (0.01) (28d.f = 0.463)

Q. Of the three associations measured which is the strongest?

Q. Would you say that there is any graphical evidence of 'clustering'? (This refers to the possibility of patterns or 'clouds' of data points appearing in a 2D or 3D chart).

Task: Produce and comment upon the findings for planes numbered 7, 12 & 27 from the chart above. For example, which of these planes would you say was the most efficient in terms of maintenance costs vs distance travelled.

 

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A more complex Correlation matrix....

Here is a data set about the value of cars. There are 4 different parametric variables involved so there will be 6 possible rp values to calculate. 10 motor cars, all registered in October 2000, were compared. The 4 variables were:

A	Mileage at 3 years old
B	Original price
C	Value at 3 years old
D	Predicted value at 5 years old

Use dataset: SPex 74 Car values

 

Here is the SPSS output:

There is a strong positive correlation between and the value at both 3 (.855) and 5 years (.948) old.
There is a fairly strong negative correlation between mileage and value at 3 (-.690) and 5 years (-.685) old.

Here is just one of the possible chart outputs......

rp = -0.690

Q. Is this what you would have expected?

There is another 'family' of statistical techniques known collectively as the 'Analysis of variance'. They are used when you want to find out about differences between a number of groups but all in one process rather than as a number of 'mini-comparisons'. They derive in part from t-tests(Focus 6 and 6a). We will explore the subject in Focus 12 (Analysis of Variance).