The relationship between two measurements

Two questions, two tools

A plant breeder looks at the link between plant height and yield; an agronomist at the link between nitrogen rate and yield. Two different questions can be asked:

• Do they change together, and how strongly? This is the question of correlation. Both variables have equal status; nobody asks which affects which.
• If I know one, how well can I predict the other? This is the question of regression. One variable is independent and controlled (nitrogen rate); the other is dependent and predicted from it (yield).

The correlation coefficient (r)

$r$ is a number between −1 and +1. Its sign shows the direction of the relationship and its size the strength:

r	Meaning	Example from the field
Close to +1	Strong positive: as one rises, the other rises steadily	Spike length and kernels per spike
Around 0	No linear relationship	Plot number and yield
Close to −1	Strong negative: as one rises, the other falls steadily	Weed density and yield

As a rough rule, $|r| < 0.3$ is weak, $0.3$–$0.7$ moderate and above $0.7$ strong. But even a large $r$ calculated from few points can arise by chance, so whether $r$ is significant is checked separately with a t test.

The regression line

Regression finds the line through the points that minimizes the sum of the squared vertical distances to the points (the least squares method):

$$\hat{y} = a + b\,x$$

The most useful number is the slope ($b$): it tells you how much Y changes on average when X increases by one unit. In the example below the slope is about 8.45, so each extra kg of nitrogen raises yield by 8.45 kg/da on average.

The coefficient of determination ($R^2$) shows how much of the variation in Y is explained by X. $R^2 = 0.94$ means 94% of the differences in yield are explained by differences in nitrogen rate, and the remaining 6% come from other factors (soil, measurement error).

Question: Do two measurements change together? Knowing one, how well can the other be predicted?
Example: Yield (kg/da) at different nitrogen rates (kg/da). Is there a relationship between nitrogen and yield, and how much does yield rise as the rate increases?

Name of X

X values (independent variable)0, 3, 6, 9, 12, 15, 18

n = 7, mean 9

Name of Y

Y values (dependent variable)285, 320, 362, 390, 415, 428, 432

n = 7, mean 376

Significance level (α)5% (0.05)1% (0.01)10% (0.10)X to predict

Dots are the measurements, the line is the regression line. The green dot is the prediction for your X.

Regression equation

$\hat{y} = 299.93 + 8.452\,x$

Correlation (r)

0.969

Determination (R²)

93.9%

Prediction for x = 10

384.45

When X increases by one unit, Y increases by 8.452 units on average. 93.9% of the variation in Y is explained by X; the relationship is strong.

1. Slope (b): how much Y changes on average when X increases by one unit.

   $$b = \dfrac{S_{xy}}{S_{xx}} = \dfrac{2130}{252} = 8.4524$$

2. Intercept (a): makes the line pass through the mean point (x̄, ȳ).

   $$a = \bar{y} - b\,\bar{x} = 376 - 8.4524 \cdot 9 = 299.9286$$

3. Regression equation:

   $$\hat{y} = 299.929 + 8.4524\,x$$

4. Coefficient of determination: the share of the variation in Y explained by X.

   $$R^2 = r^2 = 0.9392 \;(93.9\%)$$

Is the relationship significant?

$H_0: \rho = 0$$H_1: \rho \neq 0$

The red area is the rejection region (α = 0.05). The green line is the value calculated from your data.

H₀ rejected: the difference is significant

There is a strong, positive (as one increases, so does the other) and statistically significant relationship between X and Y.

p-value = 0.0003 (significant at 1%, **)

Correlation is not causation: two variables changing together does not mean one causes the other.

How was it calculated?

$t_{calc}$ = 8.785

1. Compute the corrected sums of squares and products.

   $$S_{xx} = \sum x^2 - \dfrac{(\sum x)^2}{n} = 252$$
2. $$S_{yy} = \sum y^2 - \dfrac{(\sum y)^2}{n} = 19170$$
3. $$S_{xy} = \sum xy - \dfrac{\sum x \sum y}{n} = 2130$$

4. Correlation coefficient: the direction and strength of the relationship (between −1 and +1).

   $$r = \dfrac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}} = \dfrac{2130}{\sqrt{252 \cdot 19170}} = 0.9691$$

5. A t test checks whether r differs from zero.

   $$t = \dfrac{r\sqrt{n-2}}{\sqrt{1-r^2}} = \dfrac{0.9691\sqrt{5}}{\sqrt{1-0.9392}} = 8.785$$

6. Compare the calculated value with the table (critical) value.

   $$|t_{calc}| = 8.785 > t_{tab} = 2.571$$

7. The calculated value is in the rejection region: H₀ is rejected. The p-value (0.0003) is smaller than α = 0.05.

Don’t trust the line too much

Not every relationship is linear. Look closely at the points in the example: as the nitrogen rate rises, the yield gain slows down and almost stops between 15 and 18 kg. In reality yield reaches a plateau at some point and even falls at excessive rates. The line summarizes this curve roughly but overstates the gain at high rates. Don’t assume a relationship is really linear just because $R^2$ is high; always look at the scatter plot first.

Don’t go outside the range you measured. Predicting yield at 40 kg of nitrogen with a line measured between 0 and 18 kg means extending the line into a region it has never seen. The calculator warns you when you ask for such a prediction.