3:10 - 3:30 | Check-In and Announcements / Exam Debrief.
3:30 - 4:15 | The Linear Model (PREDICTION IS A LINE)
4:15 - 4:30 | Break Time
4:30 - 5:00 | The Linear Model (ERROR IN PREDICTIONS)
5:00 - 5:30 | Reliability and Validity [RECAP]
5:30 - 6:00 | Final Project Workshop (Milestone #3)
Previously, we discussed how the mean could be used to make predictions of individuals.
\(\huge y_i = \hat{Y} + \epsilon_i\)
\(\Large y_i\) = the DV = the individual’s actual score we are trying to predict (remember \(_i\) = index; a specific individual.)
\(\Large \hat{Y}\) = our prediction (the mean).
\(\Large \epsilon\) = residual error = distance between the predicted values of y and the individual’s actual value of y
d <- read.csv("~/Dropbox/!WHY STATS/Class Datasets/101 - Class Datasets - FA25/mini_cal_data.csv", stringsAsFactors = T)
plot(d$insta.follows, main = "Mean as a Model (Red Line)",
xlab = "Index (Row in Dataset)",
ylab = "# Of Accounts a Person Follows")
abline(h = mean(d$insta.follows, na.rm = T), lwd = 5, col = 'red')
We also talked about how we could quantify the total error in these predictions, by adding up the squared residual errors (the sum of squared errors).
residual <- d$insta.follows - mean(d$insta.follows, na.rm = T)
SST <- sum(residual^2, na.rm = T)
SST[1] 70644151This number made no sense, but it is a critical statistic, since it quantifies how valid our predictions of individuals were when using the mean to make predictions.
To give the statistics some context, we divided the sum of squared errors by the sample size (this is the variance) and then un-squared this number (by taking the square root). This new statistic - the standard deviation - served as an average of residual error that describes how far the average person differs from the mean.
n <- length(na.omit(d$insta.follows)) # total number of individuals; omitting missing data.
sqrt(SST/(n-1)) # the equation for the standard deviation[1] 598.832sd(d$insta.follows, na.rm = T) # the function to get the same answer.[1] 598.832As scientists, our goal is to make accurate predictions of individuals. So we would want to find a way to make the sum of squared errors equal zero - have no error in our predictions. The mean is a good starting place, but it’s one number. And people are complex.
| the mean | the linear model ™ © |
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The mean is an okay starting place for our predictions, but we can try to do better!
DISCUSS :
ICE BREAKER : if you had to live inside one social media platform, what would it be and why???
THINK ABOUT A LINEAR MODEL : how do you think the variables (above) would help (or not help) us predict the number of accounts someone follows on instagram (insta.follows)? Why / why not???
names(d[,sapply(d, is.numeric)]) [1] "pace" "engage" "fb.friends" "insta.followers"
[5] "insta.follows" "bored" "thirsty" "tired"
[9] "satlife" "oski.love" "r.love" "socmeduse"
[13] "attention" "hrs.sleep" "data.power" "corp.power"
[17] "success.work" "success.priv" "selfes" "shoe.size"
[21] "height" "is.happy" "number.pets" | Variables we think would help us make predictions | Variables we think would not help us make predictions |
The model is a line that updates our predictions of one variable based on knowledge of another.
Define your model : what is your DV? What are your IVs? How do you think they will be related???
Graph your DV and IV(s) : make sure the data look good.
par(mfrow = c(1,2))
hist(d$insta.follows)
hist(d$socmeduse)
plot(insta.follows ~ socmeduse, data = d)
mod <- lm(insta.follows ~ socmeduse, data = d) # defines the model; saves as mod
plot(insta.follows ~ socmeduse, data = d) # graphs the relationship.
abline(mod, lwd = 5, col = 'red') # draws a red line of width five based on mod
coef(mod) # shows us the terms inside mod.(Intercept) socmeduse
96.64377 86.65467 equation for a line : y = a + bX
\(\Large y_i\) = the DV = each individual’s actual score on the dependent variable.
\(\Large a\) = the intercept = the starting place for our prediction. You can think of the intercept as “the predicted value of y when all x values are zero”.)
\(\Large X_i\) = the IV = the individual’s actual score on the independent variable (a different variable than the DV).
\(\Large b_1\) = the slope = an adjustment we make in our prediction of y, based on the individual’s x value.
\(\Large \epsilon_i\) = residual error = the distance between our prediction and the individual’s actual y value.
## Student Examples Go Here?
par(mfrow = c(1,2))
plot(d$insta.follows, main = "Using the Mean To Make Predictions (Black Line)",
xlab = "Index (Row in Dataset)",
ylab = "# Of Accounts a Person Follows")
abline(h = mean(d$insta.follows, na.rm = T), lwd = 5, col = 'black')
plot(insta.follows ~ socmeduse, data = d,
main = "Using Linear Model to Make Predictions") # graphs the relationship
abline(mod, lwd = 5, col = 'red') # draws a red line of width five based on mod
residual <- d$insta.follows - mean(d$insta.follows, na.rm = T)
SST <- sum(residual^2, na.rm = T)
SST[1] 70644151head(mod$residuals) # the residuals from my model 1 2 3 4 5 6
-96.64377 -637.22645 -316.57178 -583.22645 -743.19046 13.46421 sum(mod$residuals) # add up to zero[1] -3.637979e-12SSM <- sum(mod$residuals^2)
SSM[1] 63102736SST - SSM[1] 7541415(SST - SSM)/SST[1] 0.1067522Relevance to Psychological Science
Relevance to Real-Life.
1 Brandt, D. J., Sommer, J., Krach, S., Bedenbender, J., Kircher, T., Paulus, F. M., & Jansen, A. (2013). Test-retest reliability of fMRI brain activity during memory encoding. Frontiers in psychiatry, 4, 163. [Link to Full Article]
2 Bennett, C. M., Miller, M. B., & Wolford, G. L. (2009). Neural correlates of interspecies perspective taking in the post-mortem Atlantic Salmon: an argument for multiple comparisons correction. Neuroimage, 47(Suppl 1), S125. [Link to Full Article]
How would you evaluate the reliability and validity of the STEP COUNTER on your phone???
| Term | Way of Testing |
| face : does our measure or result look like what it should look like? | |
| convergent : is our measure similar to related concepts? | |
| discriminant : is our measure different from unrelated concepts? | |
| test-retest : do we get the same result if we take multiple measures? | |
| interrater reliability : would another observer make the same measurements? | |
| inter-item reliability : would one item in the likert scale be related to others? |
