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I'm giving a presentation about fitting lines. I have a simple linear function, $y=1x+b$. I'm trying to get scattered data points that I can put in a scatter plot that will keep my line of best fit the same equation.

I'd love to learn this technique in either R or Excel--whichever is easier.

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  • 1
    $\begingroup$ The multiple regression case with any set of coefficients (of which yours is a special case) is discussed in item (2) of this answer. Following the steps there resolves the simple regression case. The approach works in just about any package where you can simulate random values of the desired distribution, and fit regression models. $\endgroup$ – Glen_b May 12 '15 at 22:22
  • $\begingroup$ autodeskresearch.com/publications/samestats presents a nice generalization of this: simulated annealing is used to create scatterplots that not only have desired values of summary stats, but they also have a determined shape (such as the "datasaurus"). This is work by Justin Matejka and George Fitzmaurice entitled Same Stats, Different Graphs: Generating Datasets with Varied Appearance and Identical Statistics through Simulated Annealing. $\endgroup$ – whuber May 3 '17 at 15:42
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Pick any $(x_i)$ provided at least two of them differ. Set an intercept $\beta_0$ and slope $\beta_1$ and define

$$y_{0i} = \beta_0 + \beta_1 x_i.$$

This fit is perfect. Without changing the fit, you can modify $y_0$ to $y = y_0 + \varepsilon$ by adding any error vector $\varepsilon=(\varepsilon_i)$ to it provided it is orthogonal both to the vector $x = (x_i)$ and the constant vector $(1,1,\ldots, 1)$. An easy way to obtain such an error is to pick any vector $e$ and let $\varepsilon$ be the residuals upon regressing $e$ against $x$. In the code below, $e$ is generated as a set of independent random normal values with mean $0$ and common standard deviation.

Furthermore, you can even preselect the amount of scatter, perhaps by stipulating what $R^2$ should be. Letting $\tau^2 = \text{var}(y_i) = \beta_1^2 \text{var}(x_i)$, rescale those residuals to have a variance of

$$\sigma^2 = \tau^2\left(1/R^2 - 1\right).$$

This method is fully general: all possible examples (for a given set of $x_i$) can be created in this way.


Examples

Anscombe's Quartet

We can easily reproduce Anscombe's Quartet of four qualitatively distinct bivariate datasets having the same descriptive statistics (through second order).

Figure

The code is remarkably simple and flexible.

set.seed(17)
rho <- 0.816                                             # Common correlation coefficient
x.0 <- 4:14
peak <- 10
n <- length(x.0)

# -- Describe a collection of datasets.
x <- list(x.0, x.0, x.0, c(rep(8, n-1), 19))             # x-values
e <- list(rnorm(n), -(x.0-peak)^2, 1:n==peak, rnorm(n))  # residual patterns
f <- function(x) 3 + x/2                                 # Common regression line

par(mfrow=c(2,2))
xlim <- range(as.vector(x))
ylim <- f(xlim + c(-2,2))
s <- sapply(1:4, function(i) {
  # -- Create data.
  y <- f(x[[i]])                                         # Model values
  sigma <- sqrt(var(y) * (1 / rho^2 - 1))                # Conditional S.D.
  y <- y + sigma * scale(residuals(lm(e[[i]] ~ x[[i]]))) # Observed values

  # -- Plot them and their OLS fit.
  plot(x[[i]], y, xlim=xlim, ylim=ylim, pch=16, col="Orange", xlab="x")
  abline(lm(y ~ x[[i]]), col="Blue")

  # -- Return some regression statistics.
  c(mean(x[[i]]), var(x[[i]]), mean(y), var(y), cor(x[[i]], y), coef(lm(y ~ x[[i]])))
})
# -- Tabulate the regression statistics from all the datasets.
rownames(s) <- c("Mean x", "Var x", "Mean y", "Var y", "Cor(x,y)", "Intercept", "Slope")
t(s)

The output gives the second-order descriptive statistics for the $(x,y)$ data for each dataset. All four lines are identical. You can easily create more examples by altering x (the x-coordinates) and e (the error patterns) at the outset.

Simulations

This R function generates vectors $y$ according to the specifications of $\beta=(\beta_0,\beta_1)$ and $R^2$ (with $0 \le R^2 \le 1$), given a set of $x$ values.

simulate <- function(x, beta, r.2) {
  sigma <- sqrt(var(x) * beta[2]^2 * (1/r.2 - 1))
  e <- residuals(lm(rnorm(length(x)) ~ x))
  return (y.0 <- beta[1] + beta[2]*x + sigma * scale(e))
}

(It wouldn't be difficult to port this to Excel--but it's a little painful.)

As an example of its use, here are four simulations of $(x,y)$ data using a common set of $60$ $x$ values, $\beta=(1,-1/2)$ (i.e., intercept $1$ and slope $-1/2$), and $R^2 = 0.5$.

Figure

n <- 60
beta <- c(1,-1/2)
r.2 <- 0.5   # Between 0 and 1

set.seed(17)
x <- rnorm(n)

par(mfrow=c(1,4))
invisible(replicate(4, {
  y <- simulate(x, beta, r.2)
  fit <- lm(y ~ x)
  plot(x, y)
  abline(fit, lwd=2, col="Red")
}))

By executing summary(fit) you can check that the estimated coefficients are exactly as specified and the multiple $R^2$ is the intended value. Other statistics, such as the regression p-value, can be adjusted by modifying the values of the $x_i$.

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