To install and load all the packages used in this chapter, run the following code:

# install packages (only if not already installed)
for (pkg in c("here", "readxl", "tidyverse")) {
  if (!require(pkg, character.only = TRUE)) install.packages(pkg)
}

# load packages
library(tidyverse)
library(here)
library(readxl)

Data

This dataset contains information from two farmers, Max and Peter, who applied different amounts of fertilizer to their crop fields and recorded the resulting increase in yield compared to unfertilized control plots¹.

Import

dat <- read_csv(
  file = here("data", "yield_increase.csv")
)

dat

Rows: 20 Columns: 3
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (1): farmer
dbl (2): fert, yield_inc

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

# A tibble: 20 × 3
   farmer  fert yield_inc
   <chr>  <dbl>     <dbl>
 1 Max        1       0.2
 2 Max        2       0.3
 3 Max        3       0.5
 4 Max        3       0.6
 5 Max        4       0.6
 6 Max        4       0.5
 7 Max        4       0.7
 8 Max        5       0.6
 9 Max        7       0.8
10 Max        8       1  
11 Peter      1       0.1
12 Peter      1       0.1
13 Peter      1       0.2
14 Peter      1       0.2
15 Peter      1       0.1
16 Peter      3       0.3
17 Peter      5       0.5
18 Peter      6       0.8
19 Peter      8       0.9
20 Peter      9       1.3

Goal

The goal of this analysis is to answer the question of how fertilizer application relates to crop yield increase. Note that we can ignore the column farmer, since we do not care whether data came from Peter or Max. Thus, we only focus on the two numeric columns fert and yield_inc. For them, we will do a correlation and a regression analysis.

Exploring

In order to explore this dataset, we could first have a quick look at the data via

summary(dat)

    farmer               fert        yield_inc    
 Length:20          Min.   :1.00   Min.   :0.100  
 Class :character   1st Qu.:1.00   1st Qu.:0.200  
 Mode  :character   Median :3.50   Median :0.500  
                    Mean   :3.85   Mean   :0.515  
                    3rd Qu.:5.25   3rd Qu.:0.725  
                    Max.   :9.00   Max.   :1.300

to learn that the amount of fertilizer applied ranges from 1 to 9 kg/ha with a mean of approximately 3.7 kg/ha, while the measured yield increases range from 0.1 to 1.3 tons/ha with a mean of about 0.5 tons/ha.

And now it is finally time to create our first ggplot. Our goal is to create it like so:

The plot shows a clear trend of increasing crop yields with higher fertilizer application - which is what we would expect.

Where is the ggplot code?!

To see and understand the code needed to create this ggplot and all other ggplots in this chapter, please go to the next chapter. You can do this right now or after reading through this chapter.

Correlation

One way of actually putting a number on this relationship is to estimate the correlation. When people talk about correlation ($\rho$ or $r$) in statistics, they usually refer to the Pearson correlation coefficient, which is a measure of linear correlation between two numeric variables. Correlation can only have values between -1 and 1, where 0 means no correlation, while all other possible values are either negative or positive correlations. The farther away from 0, the stronger is the correlation. Here are a few examples:

Simply put, a positive correlation means “if one variable gets bigger, the other also gets bigger” and a negative correlation means “if one variable gets bigger, the other gets smaller”. Therefore, it does not matter which of the two variables is the first (“x”) or the second (“y”) variable. Moreover, a correlation estimate is not like a model and it cannot make predictions. Finally, “correlation does not imply causation” means that just because you found a (strong) correlation between two things, you cannot conclude that there is a cause-and-effect relationship between the two.

Tip

Have a look at these spurious-correlations for some funny examples of correlation without causation.
Play around with this neat tool to get a better feeling for the relationshop between correlation and data.

Get it

If you only want to get the actual correlation estimate, you can use the function cor() and provide the two numeric variables (as vectors). So in our case, we can extract the column with fertilizer application from our data object dat with dat$fert and the column with yield increase with dat$yield_inc. Remember: the $ sign can be used to extract a column from a table. So the command to get the correlation between fertilizer application and yield increase looks like this:

cor(dat$fert, dat$yield_inc)

[1] 0.9559151

Accordingly, the correlation between fertilizer application and yield increase in our sample is very strong, since it is close to 1. This suggests that increasing fertilizer tends to be associated with increasing crop yield.

Test it

If you would like additional information, such as a confidence interval and a test resulting in a p-value, you can use cor.test() instead of cor().

mycor <- cor.test(dat$fert, dat$yield_inc)
mycor


    Pearson's product-moment correlation

data:  dat$fert and dat$yield_inc
t = 13.811, df = 18, p-value = 5.089e-11
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.8897837 0.9827293
sample estimates:
      cor 
0.9559151

p-values and statistical significance

The topic of hypothesis testing, p-values and statistical significance is a bit more complex and has its own chapter. You can read it now or after this chapter.

Looking at this longer output, you can see the sample estimate at the bottom, a confidence interval above it and a p-value with the corresponding test hypothesis above that. Run ?cor.test() and look at the “Details” section for more info. Here, our correlation estimate is significantly different from 0, since the p-value is much smaller than 0.05. Furthermore, the confidence interval shown means that we are 95% sure that the true correlation is somewhere in that range.

People would report this in their results section as e.g. “The correlation between fertilizer application and yield increase was 0.96 (95% CI: 0.89, 0.98) and statistically significant (p < 0.001).”

Simple linear regression

When people talk about regression in statistics, they usually refer to simple linear regression, which - simply put - finds the best straight line that goes through dots in a scatter plot of two numeric variables.

The linear model behind such a straight line is simply:

\[ y = \alpha + \beta x\]

where $\alpha$ or $a$ is the intercept and $\beta$ or $b$ is the slope, while $y$ and $x$ are our data points. Fitting such a regression is really just finding the optimal estimates for $\alpha$ and $\beta$.

In contrast to correlation, a simple linear regression is a model and it therefore matters which variable is $y$ (dependent variable) and which is $x$ (independent), because after fitting the regression, the latter can be used to predict the former.

Tip

Visit this website, type in “y=a+bx” in the box in the upper left corner, hit Enter and then play around with the values of $a$ and $b$ in the boxes below. You can see how changing the slope and intercept changes the line. The slope ($b$) indicates how much $y$ increases when $x$ increases by 1 unit, while the intercept ($a$) indicates the expected value of $y$ when $x$ is 0.

Get it

In R, we can use the lm() function for fitting linear models so that it fits the simple linear regression equation shown above easily:

reg <- lm(formula = yield_inc ~ fert,
          data = dat)

As you can see, we refer to our data object dat in the data = argument so that in the formula = argument we only need to write the names of the respective columns in dat. Furthermore, we store the results in the reg object. When looking at this object, we get the following results:

reg


Call:
lm(formula = yield_inc ~ fert, data = dat)

Coefficients:
(Intercept)         fert  
    0.04896      0.12105

First, our command is repeated and then the “Coefficients” are shown, which are indeed the estimates for $a$ and $b$. So the best straight line is:

\[ yield\_inc = 0.049 + 0.121 * fert \]

which looks like this:

Here is a little more info why formula = yield_inc ~ fert leads to R estimating the $a$ and $b$ we want: What makes sense is that yield_inc is $y$, fert is $x$ and ~ would therefore be the $=$ in our equation. However, why is it we never had to write anything about $a$ or $b$? The answer is that (i) when fitting a linear model, there is usually always an intercept (=$a$) by default and (ii) when writing a numeric variable (=fert) as on the right side of the equation, it will automatically be assumed to have a slope (=$b$) multiplied with it. Accordingly, yield_inc ~ fert automatically translates to yield_inc = a + b*fert so to speak.

Is this right?

After fitting a model, you may use it to make predictions. Here is one way of obtaining the expected yield increase for applying 0 to 10 kg/ha of fertilizer according to our simple linear regression:

preddat <- tibble(fert = seq(0, 10))
preddat %>%
  mutate(predicted_yield_inc = predict(reg, newdata = preddat))

# A tibble: 11 × 2
    fert predicted_yield_inc
   <int>               <dbl>
 1     0              0.0490
 2     1              0.170 
 3     2              0.291 
 4     3              0.412 
 5     4              0.533 
 6     5              0.654 
 7     6              0.775 
 8     7              0.896 
 9     8              1.02  
10     9              1.14  
11    10              1.26

You may notice that according to our model, the expected yield increase when applying 0 kg/ha of fertilizer is actually 0.049 tons/ha and thus larger than 0. This is unexpected. When no additional fertilizer is applied, there should be no additional yield compared to the unfertilized control plots. So what went wrong?

First of all, data will never be perfect. Even if the true value for something is 0, its estimate based on measured data will never be exactly 0.000000… . Instead, there is always “noise” in the data, e.g. measurement errors: The farmers may have miscalculated the exact amount of fertilizer or there might be errors in measuring the yield increase or random environmental influences etc.

So I would like you to think about the issue from two other angles:

Are the results really saying the intercept is > 0?
Did we even ask the right question or should we have fitted a different model?

Are the results really saying the intercept is > 0?

No, they are not. Yes, the sample estimate for the intercept is 0.049, but when looking at more detailed information via e.g. summary() we can see more:

summary(reg)


Call:
lm(formula = yield_inc ~ fert, data = dat)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.154206 -0.070011 -0.004206  0.039202  0.187891 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.048963   0.040592   1.206    0.243    
fert        0.121049   0.008764  13.811 5.09e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.1009 on 18 degrees of freedom
Multiple R-squared:  0.9138,    Adjusted R-squared:  0.909 
F-statistic: 190.8 on 1 and 18 DF,  p-value: 5.089e-11

You can see that the p-value for the intercept is larger than 0.05 and thus saying that we could not find the intercept to be significantly different from 0 (see chapter “033_tests_and_pvalues” for details on the interpretation of this).

Should we have fitted a different model?

We certainly could have and we will actually do it now. It must be clear that statistically speaking there was nothing wrong with our analysis. However, from an agronomic standpoint or in other words - because of our background knowledge and expertise as agricultural scientists - we could have indeed actively decided for a regression analysis that does not have an intercept and is thus forced to start at 0 in terms of yield increase. After all, statistics is just a tool to help us make conclusions. It is a powerful tool, but it will always be our responsibility to “ask the right questions” i.e. apply expedient methods.

A simple linear regression without an intercept is strictly speaking no longer “simple”, since it no longer has the typical equation, but instead this one:

\[ y = \beta x\]

To tell lm() that it should not estimate the default intercept, we simply add 0 + right after the ~. As expected, we only get one estimate for the slope:

reg_noint <- lm(formula = yield_inc ~ 0 + fert, data = dat)
reg_noint


Call:
lm(formula = yield_inc ~ 0 + fert, data = dat)

Coefficients:
  fert  
0.1298

meaning that this regression with no intercept is estimated as

\[ yield\_inc = 0.1298 * fert \]

and must definitely predict 0 yield_inc when applying 0 kg/ha of fert. As a final result, we can compare both regression lines visually in a ggplot:

Tip

Finally, here is a little tip for you: The package {broom} is a very useful package for tidying up the output of models or other statistical analyses. It is not built-in, so you need to install it first with install.packages("broom") and load it with library(broom). You can then use its three functions tidy(), glance() and augment() to get the results of your statistical analysis in a “tidy” tibble format. This is especially useful if your goal is to export those results. For example, we can use tidy() on both our correlation and linear regression results:

library(broom)
tidy(mycor)

# A tibble: 1 × 8
  estimate statistic  p.value parameter conf.low conf.high method    alternative
     <dbl>     <dbl>    <dbl>     <int>    <dbl>     <dbl> <chr>     <chr>      
1    0.956      13.8 5.09e-11        18    0.890     0.983 Pearson'… two.sided

tidy(reg)

# A tibble: 2 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)   0.0490   0.0406       1.21 2.43e- 1
2 fert          0.121    0.00876     13.8  5.09e-11

Try running the code yourself and compare this to just running mycor and reg without the tidy() function. A list of all things that can be tidied up with this function is given here

Wrapping Up

Congratulations! You’ve learned the fundamentals of correlation and regression analysis, two of the most commonly used statistical techniques for analyzing relationships between numeric variables in agricultural research.

Key Takeaways

Correlation measures the strength and direction of a linear relationship between two variables:
- Values range from -1 to 1
- Closer to 1: Strong positive correlation
- Closer to -1: Strong negative correlation
- Near 0: Little to no correlation
- Correlation does not imply causation
Simple Linear Regression fits a straight line to data to model the relationship between variables:
- Formula: y = α + βx (with intercept) or y = βx (without intercept)
- Here, the slope (β) indicates how much yield increases when fertilizer application increases by 1 kg/ha
- Here, the intercept (α) indicates the expected yield increase when no fertilizer is applied
- Enables predictions of yield increases based on planned fertilizer applications
Model Evaluation is crucial for determining if your regression makes sense:
- Check if coefficients are statistically significant
Statistical vs. Practical Significance
- Sometimes domain knowledge (like knowing yield increase should be zero with zero fertilizer) may suggest constraints on what the model should look like.
- Remember that statistical tools are guides, but your expertise should inform your final interpretation and modeling choices.

Footnotes

Numbers for applied fertilizer in kg/ha and yield increase in t/ha are made up and chosen to be simple instead of realistic.↩︎

Citation

BibTeX citation:

@online{schmidt2026,
  author = {{Dr. Paul Schmidt}},
  publisher = {BioMath GmbH},
  title = {4. {Correlation} \& {Regression}},
  date = {2026-02-07},
  url = {https://biomathcontent.netlify.app/content/r_intro/04_correlationregression.html},
  langid = {en}
}

For attribution, please cite this work as:

Dr. Paul Schmidt. 2026. “4. Correlation & Regression.” BioMath GmbH. February 7, 2026. https://biomathcontent.netlify.app/content/r_intro/04_correlationregression.html.

--- title: "4. Correlation & Regression" subtitle: "Understanding relationships between numeric variables" --- To install and load all the packages used in this chapter, run the following code: ```{r} #| label: setup-packages #| output: false #| messages: false #| warning: false #| results: 'hide' # install packages (only if not already installed) for (pkg in c("here", "readxl", "tidyverse")) { if (!require(pkg, character.only = TRUE)) install.packages(pkg) } # load packages library(tidyverse) library(here) library(readxl) ``` # Data This dataset contains information from two farmers, Max and Peter, who applied different amounts of fertilizer to their crop fields and recorded the resulting increase in yield compared to unfertilized control plots[^1]. [^1]: Numbers for applied fertilizer in kg/ha and yield increase in t/ha are made up and chosen to be simple instead of realistic. ## Import ```{r} #| label: import-data-example #| eval: false dat <- read_csv( file = here("data", "yield_increase.csv") ) dat ``` ```{r} #| label: import-data-actual #| echo: false dat <- read_csv( file = here("example_project", "data", "yield_increase.csv"), ) dat ``` ## Goal The goal of this analysis is to answer the question of how fertilizer application relates to crop yield increase. Note that we can ignore the column `farmer`, since we do not care whether data came from Peter or Max. Thus, we only focus on the two *numeric* columns `fert` and `yield_inc`. For them, we will do a correlation and a regression analysis. ## Exploring In order to explore this dataset, we could first have a quick look at the data via ```{r} #| label: data-summary summary(dat) ``` to learn that the amount of fertilizer applied ranges from 1 to 9 kg/ha with a mean of approximately 3.7 kg/ha, while the measured yield increases range from 0.1 to 1.3 tons/ha with a mean of about 0.5 tons/ha. And now it is finally time to create our first ggplot. Our goal is to create it like so: ```{r} #| label: initial-scatterplot #| echo: false ggplot(data = dat) + aes(x = fert, y = yield_inc) + geom_point(size = 2) + scale_x_continuous( name = "Fertilizer applied (kg/ha)", limits = c(0, 10), breaks = seq(0, 10, by = 1), expand = expansion(mult = c(0, 0.1)) ) + scale_y_continuous( name = "Yield increase (tons/ha)", limits = c(0, NA), expand = expansion(mult = c(0, 0.1)) ) + theme_classic() + labs(title = "Plot A") ``` The plot shows a clear trend of increasing crop yields with higher fertilizer application - which is what we would expect. :::{.callout-note title="Where is the ggplot code?!"} To see and understand the code needed to create this ggplot and all other ggplots in this chapter, please go to the next chapter. You can do this right now or after reading through this chapter. ::: \newpage # Correlation One way of actually putting a number on this relationship is to estimate the correlation. When people talk about correlation ($\rho$ or $r$) in statistics, they usually refer to the [Pearson correlation coefficient](https://www.wikiwand.com/en/articles/Pearson_correlation_coefficient), which is a measure of linear correlation between two numeric variables. Correlation can only have values between -1 and 1, where 0 means *no correlation*, while all other possible values are either negative or positive correlations. The farther away from 0, the stronger is the correlation. Here are a few examples: ![](../../img/correlation.PNG) Simply put, a positive correlation means *"if one variable gets bigger, the other also gets bigger"* and a negative correlation means *"if one variable gets bigger, the other gets smaller"*. Therefore, it does not matter which of the two variables is the first ("x") or the second ("y") variable. Moreover, a correlation estimate is not like a model and it cannot make predictions. Finally, *"correlation does not imply causation"* means that just because you found a (strong) correlation between two things, you cannot conclude that there is a cause-and-effect relationship between the two. :::{.callout-tip title="Tip"} - Have a look at these [spurious-correlations](https://www.tylervigen.com/spurious-correlations) for some funny examples of correlation without causation. - Play around with [this neat tool](https://rpsychologist.com/correlation/) to get a better feeling for the relationshop between correlation and data. ::: ## Get it If you only want to get the actual correlation estimate, you can use the function `cor()` and provide the two numeric variables (as vectors). So in our case, we can extract the column with fertilizer application from our data object `dat` with `dat$fert` and the column with yield increase with `dat$yield_inc`. Remember: the `$` sign can be used to extract a column from a table. So the command to get the correlation between fertilizer application and yield increase looks like this: ```{r} #| label: correlation-estimate cor(dat$fert, dat$yield_inc) ``` Accordingly, the correlation between fertilizer application and yield increase in our sample is very strong, since it is close to 1. This suggests that increasing fertilizer tends to be associated with increasing crop yield. ## Test it If you would like additional information, such as a confidence interval and a test resulting in a p-value, you can use `cor.test()` instead of `cor()`. ```{r} #| label: correlation-test mycor <- cor.test(dat$fert, dat$yield_inc) mycor ``` :::{.callout-note title="p-values and statistical significance"} The topic of hypothesis testing, p-values and statistical significance is a bit more complex and has its own chapter. You can read it now or after this chapter. ::: Looking at this longer output, you can see the sample estimate at the bottom, a confidence interval above it and a p-value with the corresponding test hypothesis above that. Run `?cor.test()` and look at the "Details" section for more info. Here, our correlation estimate is significantly different from 0, since the p-value is much smaller than 0.05. Furthermore, the confidence interval shown means that we are 95% sure that the true correlation is somewhere in that range. People would report this in their results section as e.g. "The correlation between fertilizer application and yield increase was 0.96 (95% CI: 0.89, 0.98) and statistically significant (p < 0.001)." \newpage # Simple linear regression When people talk about regression in statistics, they usually refer to simple linear regression, which - simply put - finds the best straight line that goes through dots in a scatter plot of two numeric variables. The linear model behind such a straight line is simply: $$ y = \alpha + \beta x$$ where $\alpha$ or $a$ is the intercept and $\beta$ or $b$ is the slope, while $y$ and $x$ are our data points. Fitting such a regression is really just finding the optimal estimates for $\alpha$ and $\beta$. In contrast to correlation, a simple linear regression is a model and it therefore matters which variable is $y$ (dependent variable) and which is $x$ (independent), because after fitting the regression, the latter can be used to predict the former. :::{.callout-tip title="Tip"} Visit [this website](https://www.desmos.com/calculator), type in "y=a+bx" in the box in the upper left corner, hit Enter and then play around with the values of $a$ and $b$ in the boxes below. You can see how changing the slope and intercept changes the line. The slope ($b$) indicates how much $y$ increases when $x$ increases by 1 unit, while the intercept ($a$) indicates the expected value of $y$ when $x$ is 0. ::: ## Get it In R, we can use the `lm()` function for fitting linear models so that it fits the simple linear regression equation shown above easily: ```{r} #| label: linear-model-fit reg <- lm(formula = yield_inc ~ fert, data = dat) ``` As you can see, we refer to our data object `dat` in the `data =` argument so that in the `formula =` argument we only need to write the names of the respective columns in `dat`. Furthermore, we store the results in the `reg` object. When looking at this object, we get the following results: ```{r} #| label: linear-model-coefficients reg ``` First, our command is repeated and then the "Coefficients" are shown, which are indeed the estimates for $a$ and $b$. So the best straight line is: $$ yield\_inc = 0.049 + 0.121 * fert $$ which looks like this: ```{r} #| label: scatterplot-with-regression #| echo: false ggplot(data = dat) + aes(x = fert, y = yield_inc) + geom_point(size = 2) + geom_abline( intercept = reg$coefficients[1], slope = reg$coefficients[2], color = "#00923f", linewidth = 1 ) + scale_x_continuous( name = "Fertilizer applied (kg/ha)", limits = c(0, 10), breaks = seq(0, 10, by = 1), expand = expansion(mult = c(0, 0.1)) ) + scale_y_continuous( name = "Yield increase (tons/ha)", limits = c(0, NA), expand = expansion(mult = c(0, 0.1)) ) + theme_classic() + labs(title = "Plot B") ``` Here is a little more info why `formula = yield_inc ~ fert` leads to R estimating the $a$ and $b$ we want: What makes sense is that `yield_inc` is $y$, `fert` is $x$ and `~` would therefore be the $=$ in our equation. However, why is it we never had to write anything about $a$ or $b$? The answer is that (i) when fitting a linear model, there is usually always an intercept (=$a$) by default and (ii) when writing a numeric variable (=`fert`) as on the right side of the equation, it will automatically be assumed to have a slope (=$b$) multiplied with it. Accordingly, `yield_inc ~ fert` automatically translates to `yield_inc = a + b*fert` so to speak. ## Is this right? After fitting a model, you may use it to make predictions. Here is one way of obtaining the expected yield increase for applying 0 to 10 kg/ha of fertilizer according to our simple linear regression: ```{r} #| label: model-predictions preddat <- tibble(fert = seq(0, 10)) preddat %>% mutate(predicted_yield_inc = predict(reg, newdata = preddat)) ``` You may notice that according to our model, the expected yield increase when applying 0 kg/ha of fertilizer is actually 0.049 tons/ha and thus larger than 0. This is unexpected. When no additional fertilizer is applied, there should be no additional yield compared to the unfertilized control plots. So what went wrong? First of all, data will never be perfect. Even if the true value for something is 0, its estimate based on measured data will never be exactly 0.000000... . Instead, there is always "noise" in the data, e.g. measurement errors: The farmers may have miscalculated the exact amount of fertilizer or there might be errors in measuring the yield increase or random environmental influences etc. So I would like you to think about the issue from two other angles: 1. Are the results really saying the intercept is \> 0? 2. Did we even ask the right question or should we have fitted a different model? ### Are the results really saying the intercept is \> 0? No, they are not. Yes, the sample estimate for the intercept is 0.049, but when looking at more detailed information via e.g. `summary()` we can see more: ```{r} #| label: model-summary summary(reg) ``` You can see that the p-value for the intercept is larger than 0.05 and thus saying that we could not find the intercept to be significantly different from 0 (see chapter "033_tests_and_pvalues" for details on the interpretation of this). ### Should we have fitted a different model? We certainly **could** have and we will actually do it now. It must be clear that statistically speaking there was nothing wrong with our analysis. However, from an agronomic standpoint or in other words - because of our background knowledge and expertise as agricultural scientists - we could have indeed actively decided for a regression analysis that does **not** have an intercept and is thus forced to start at 0 in terms of yield increase. After all, statistics is just a tool to help us make conclusions. It is a powerful tool, but it will always be our responsibility to "ask the right questions" i.e. apply expedient methods. A simple linear regression without an intercept is strictly speaking no longer "simple", since it no longer has the typical equation, but instead this one: $$ y = \beta x$$ To tell `lm()` that it should not estimate the default intercept, we simply add `0 +` right after the `~`. As expected, we only get one estimate for the slope: ```{r} #| label: no-intercept-model reg_noint <- lm(formula = yield_inc ~ 0 + fert, data = dat) reg_noint ``` meaning that this regression with no intercept is estimated as $$ yield\_inc = 0.1298 * fert $$ and must definitely predict 0 `yield_inc` when applying 0 kg/ha of `fert`. As a final result, we can compare both regression lines visually in a ggplot: ```{r} #| label: compare-regression-lines #| echo: false ggplot(data = dat) + aes(x = fert, y = yield_inc) + geom_point(size = 2) + geom_abline( intercept = reg$coefficients[1], slope = reg$coefficients[2], color = "#00923f", linewidth = 1 ) + geom_abline( intercept = 0, slope = reg_noint$coefficients[1], color = "#e4572e", linewidth = 1 ) + scale_x_continuous( name = "Fertilizer applied (kg/ha)", limits = c(0, 10), breaks = seq(0, 10, by = 1), expand = expansion(mult = c(0, 0.1)) ) + scale_y_continuous( name = "Yield increase (tons/ha)", limits = c(0, NA), expand = expansion(mult = c(0, 0.1)) ) + theme_classic() + labs(title = "Plot C") ``` \newpage :::{.callout-tip title="Tip"} Finally, here is a little tip for you: The [package `{broom}`](https://broom.tidymodels.org/index.html) is a very useful package for tidying up the output of models or other statistical analyses. It is not built-in, so you need to install it first with `install.packages("broom")` and load it with `library(broom)`. You can then use its three functions `tidy()`, `glance()` and `augment()` to get the results of your statistical analysis in a "tidy" tibble format. This is especially useful if your goal is to export those results. For example, we can use `tidy()` on both our correlation and linear regression results: ```{r} #| label: broom-tidy-correlation library(broom) tidy(mycor) ``` ```{r} #| label: broom-tidy-regression tidy(reg) ``` Try running the code yourself and compare this to just running `mycor` and `reg` without the `tidy()` function. A list of all things that can be tidied up with this function is given [here](https://broom.tidymodels.org/articles/available-methods.html) ::: # Wrapping Up Congratulations! You've learned the fundamentals of correlation and regression analysis, two of the most commonly used statistical techniques for analyzing relationships between numeric variables in agricultural research. :::{.callout-note title="Key Takeaways"} 1. **Correlation** measures the strength and direction of a linear relationship between two variables: - Values range from -1 to 1 - Closer to 1: Strong positive correlation - Closer to -1: Strong negative correlation - Near 0: Little to no correlation - Correlation does not imply causation 2. **Simple Linear Regression** fits a straight line to data to model the relationship between variables: - Formula: y = α + βx (with intercept) or y = βx (without intercept) - Here, the slope (β) indicates how much yield increases when fertilizer application increases by 1 kg/ha - Here, the intercept (α) indicates the expected yield increase when no fertilizer is applied - Enables predictions of yield increases based on planned fertilizer applications 3. **Model Evaluation** is crucial for determining if your regression makes sense: - Check if coefficients are statistically significant 4. **Statistical vs. Practical Significance** - Sometimes domain knowledge (like knowing yield increase should be zero with zero fertilizer) may suggest constraints on what the model should look like. - Remember that statistical tools are guides, but your expertise should inform your final interpretation and modeling choices. :::