TI-84 Plus CE and Casio fx-CG50: Confusion Matrix, Practice SAT Questions with Mometrix
Confusion Matrix
Introduction
In statistical applications, particularly in medicine, we hear about the infection rates of a disease and tests that are created to designate whether people are infected with the disease. No test, at least not any that I heard of, is 100% accurate in detecting whether a person is infected with a certain virus.
Tables can be used to summarize the accuracy of a test, measuring one of four outcomes:
true positive (TP): the person is infected with a virus and the test detects the virus
false negative (FN): the person is infected with a virus but the test fails to detect it
false positive (FP): the test states the person is infected when in reality the person does not have the virus
true negative (TN): the person is not infected and the test accurate detects the person is healthy (does not have the virus)
One of the common names for this type of table is a confusion matrix.
Two of the many measurements that can be made from a confusion matrix are called sensitivity and specificity.
Sensitivity is the ratio of true positive results against all of the population that is infected.
Sensitivity = true positive / (true positive + false negative)
Specificity is the ratio of true negative results against all fo the population that is not infected.
Specificity = true negative / (false negative + true positive)
The program CONFUSE creates two 3 x 3 matrices (see the illustration below):
Matrix A: Theoretical confusion table. This takes into consideration the infection rate and test rate and calculates the expected values.
Matrix B: Simulated confusion table. The test uses a random number generator to simulate the chance of whether a person is infected by using the infection rate and whether a person's test is correct by using the test rate. The results will vary.
TI-84 Plus CE Program: CONFUSE
"EWS 2021-03-10"
ClrHome
DelVar [A]
DelVar [B]
{3,3}→dim([A])
{3,3}→dim([B])
Disp "CONFUSION MATRIX"
Input "POPULATION? ",N
Input "INFECTION RATE? ",C
Input "TEST RATE? ",T
N*C→[A](3,1)
N*(1-C)→[A](3,2)
[A](3,1)+[A](3,2)→[A](3,3)
[A](3,1)*T→[A](1,1)
[A](3,1)*(1-T)→[A](2,1)
[A](3,2)*(1-T)→[A](1,2)
[A](3,2)*T→[A](2,2)
[A](1,1)+[A](1,2)→[A](1,3)
[A](2,1)+[A](2,2)→[A](2,3)
For(I,1,N)
rand→R
[B](3,1)+(R≤C)→[B](3,1)
[B](3,2)+(R>C)→[B](3,2)
End
[B](3,1)+[B](3,2)→[B](3,3)
For(I,1,[B](3,1))
rand→R
[B](1,1)+(R≤T)→[B](1,1)
[B](2,1)+(R>T)→[B](2,1)
End
For(I,1,[B](3,2))
rand→R
[B](1,2)+(R>T)→[B](1,2)
[B](2,2)+(R≤T)→[B](2,2)
End
[B](1,1)+[B](1,2)→[B](1,3)
[B](2,1)+[B](2,2)→[B](2,3)
ClrHome
Disp "THEORY [A]"
Pause [A]
ClrHome
Disp "SIMULATION [B]"
Pause [B]
Disp "SENSITIVITY",[B](1,1)/[B](3,1)
Disp "SPECIFICITY",[B](2,2)/[B](3,2)
Casio fx-CG50 Program: CONFUSE
"EWS 2021-03-13"
ClrText
{3,3}->Dim Mat A
{3,3}->Dim Mat B
"CONFUSION MATRIX"
"POPULATION"?->N
"INFECTION RATE"?->C
"TEST RATE"?->T
N*C->Mat A[3,1]
N*(1-C)->Mat A[3,2]
Mat A[3,1]+Mat A[3,2]->Mat A[3,3]
Mat A[3,1]*T->Mat A[1,1]
Mat A[3,1]*(1-T)->Mat A[2,1]
Mat A[3,2]*(1-T)->Mat A[1,2]
Mat A[3,2]*T->Mat A[2,2]
Mat A[1,1]+Mat A[1,2]->Mat A[1,3]
Mat A[2,1]+Mat A[2,2]->Mat A[2,3]
For 1->I To N
Ran#->R
Mat B[3,1]+(R<=C)->Mat B[3,1]
Mat B[3,2]+(R>C)->Mat B[3,2]
Next
Mat B[3,1]+Mat B[3,2]->Mat B[3,3]
For 1->I To Mat B[3,1]
Ran#->R
Mat B[1,1]+(R<=T)->Mat B[1,1]
Mat B[2,1]+(R>T)->Mat B[2,1]
Next
For 1->I To Mat B[3,2]
Ran#->R
Mat B[1,2]+(R>T)->Mat B[1,2]
Mat B[2,2]+(R<=T)->Mat B[2,2]
Next
Mat B[1,1]+Mat B[1,2]->Mat B[1,3]
Mat B[2,1]+Mat B[2,2]->Mat B[2,3]
ClrText
"_Mat _A: THEORY" ⊿
Mat A ⊿
"_Mat _B: SIMULATION" ⊿
Mat B ⊿
"SENSITIVITY:"
Mat B[1,1]/Mat B[3,1] ⊿
"SPECIFICITY:"
Mat B[2,2]/Mat B[3,2]
Example
Population: N = 200
Infection Rate: 5% (enter 0.05)
Successful Test Rate: 80% (enter 0.80)
Theoretical Matrix (Matrix A):
[[ 8 38 46
2 152 154
10 190 200 ]]
Some simulated results (Matrix B, your results will vary):
Simulation 1:
[[ 7 40 47
1 152 153
8 192 200 ]]
Sensitivity ≈ 0.8750
Specificity ≈ 0.7917
Simulation 2:
[[ 5 34 39
1 160 161
6 194 200 ]]
Sensitivity ≈ 0.8333
Specificity ≈ 0.8247
Sources:
"Confusion Matrix" Wikipedia. Last Edited February 27, 2021. https://en.wikipedia.org/wiki/Confusion_matrix Retrieved March 9, 2021.
Texas Instruments "Webinar: Modeling as a Tool To Make Sense of the World Around Us" Presented by Gail Burrill and Tom Dick, Ph.D. https://education.ti.com/en/professional-development/teachers-and-teams/online-learning/on-demand-webinars/2021/mar-09-2021-modeling-as-a-tool-to-make-sense March 9, 2021
University of Nottingham. "Accuracy Table" https://www.nottingham.ac.uk/nursing/sonet/rlos/ebp/sensitivity_specificity/page_four.html Retrieved March 9, 2021
SAT Practice Problems with Mometrix
In 2018, I mentioned that I was going to practice some SAT questions (http://edspi31415.blogspot.com/2018/02/). If you are taking the SAT or want to practice, a place to go is Mometrix Test Preparation. Mometrix has online practice tests for reading, writing, and mathematics, as well as official Sample tests.
Check them out here: https://www.mometrix.com/academy/sat-practice-test/
Their math page which includes a free online practice test: https://www.mometrix.com/academy/sat-math-practice-test/
Many thanks to George Bigelow for the information and site.
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Eddie
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