1. The document proposes a theoretical formula scoring criterion (FSC) to simulate the effects of formula scoring on multiple-choice tests.
2. Through simulations in R, the FSC was shown to improve some examinees' observed scores so that they more closely approximated their true scores, though reliability was slightly reduced.
3. Further analysis over the next few months will evaluate the validity and usefulness of the FSC method for more accurately estimating examinees' true abilities when formula scoring is implemented.
Basic of Statistical Inference Part-IV: An Overview of Hypothesis TestingDexlab Analytics
The fourth part of the basic of statistical inference series puts its focus on discussing the concept of hypothesis testing explaining all the nuances.
Basic of Statistical Inference Part-III: The Theory of Estimation from Dexlab...Dexlab Analytics
In this 3rd segment of the basic of statistical inference series, the estimation theory, its elements, methods and characteristics have been discussed.
Basic of Statistical Inference Part-IV: An Overview of Hypothesis TestingDexlab Analytics
The fourth part of the basic of statistical inference series puts its focus on discussing the concept of hypothesis testing explaining all the nuances.
Basic of Statistical Inference Part-III: The Theory of Estimation from Dexlab...Dexlab Analytics
In this 3rd segment of the basic of statistical inference series, the estimation theory, its elements, methods and characteristics have been discussed.
In classical sampling theory, why is a model always developed on a sample even when the whole data is available? Because all the observations and variables may not be needed for developing a model, because they are not relevant for the development.
Opinion mining framework using proposed RB-bayes model for text classicationIJECEIAES
Information mining is a capable idea with incredible potential to anticipate future patterns and conduct. It alludes to the extraction of concealed information from vast data sets by utilizing procedures like factual examination, machine learning, grouping, neural systems and genetic algorithms. In naive baye’s, there exists a problem of zero likelihood. This paper proposed RB-Bayes method based on baye’s theorem for prediction to remove problem of zero likelihood. We also compare our method with few existing methods i.e. naive baye’s and SVM. We demonstrate that this technique is better than some current techniques and specifically can analyze data sets in better way. At the point when the proposed approach is tried on genuine data-sets, the outcomes got improved accuracy in most cases. RB-Bayes calculation having precision 83.333.
August 1, 2010. Design of Non-Randomized Medical Device Trials Based on Sub-Classification Using Propensity Score Quintiles, Topic Contributed Session on Medical Devices, (Greg Maislin and Donald B Rubin). Joint Statistical Meetings 2010, Vancouver Canada.
Introduction to Biostatistics. This lecture was given as a part of the Introduction to Epidemiology & Community Medicine Course given for third-year medical students.
In classical sampling theory, why is a model always developed on a sample even when the whole data is available? Because all the observations and variables may not be needed for developing a model, because they are not relevant for the development.
Opinion mining framework using proposed RB-bayes model for text classicationIJECEIAES
Information mining is a capable idea with incredible potential to anticipate future patterns and conduct. It alludes to the extraction of concealed information from vast data sets by utilizing procedures like factual examination, machine learning, grouping, neural systems and genetic algorithms. In naive baye’s, there exists a problem of zero likelihood. This paper proposed RB-Bayes method based on baye’s theorem for prediction to remove problem of zero likelihood. We also compare our method with few existing methods i.e. naive baye’s and SVM. We demonstrate that this technique is better than some current techniques and specifically can analyze data sets in better way. At the point when the proposed approach is tried on genuine data-sets, the outcomes got improved accuracy in most cases. RB-Bayes calculation having precision 83.333.
August 1, 2010. Design of Non-Randomized Medical Device Trials Based on Sub-Classification Using Propensity Score Quintiles, Topic Contributed Session on Medical Devices, (Greg Maislin and Donald B Rubin). Joint Statistical Meetings 2010, Vancouver Canada.
Introduction to Biostatistics. This lecture was given as a part of the Introduction to Epidemiology & Community Medicine Course given for third-year medical students.
this is our presentation on the starting stages to for planning to prepare to go for startup funding / small business loans / invoice loans / factoring etc etc
Hypothesis TestingThe Right HypothesisIn business, or an.docxadampcarr67227
Hypothesis Testing
The Right Hypothesis
In business, or any other discipline, once the question has been asked there must be a statement as to what will or will not occur through testing, measurement, and investigation. This process is known as formulating the right hypothesis. Broadly defined a hypothesis is a statement that the conditions under which something is being measured or evaluated holds true or does not hold true. Further, a business hypothesis is an assumption that is to be tested through market research, data mining, experimental designs, quantitative, and qualitative research endeavors. A hypothesis gives the businessperson a path to follow and specific things to look for along the road.
If the research and statistical data analysis supports and proves the hypothesis that becomes a project well done. If, however, the research data proved a modified version of the hypothesis then re-evaluation for continuation must take place. However, if the research data disproves the hypothesis then the project is usually abandoned.
Hypotheses come in two forms: the null hypothesis and the alternate hypothesis. As a student of applied business statistics you can pick up any number of business statistics textbooks and find a number of opinions on which type of hypothesis should be used in the business world. For the most part, however, and the safest, the better hypothesis to formulate on the basis of the research question asked is what is called the null hypothesis. A null hypothesis states that the research measurement data gathered will not support a difference, relationship, or effect between or amongst those variables being investigated. To the seasoned research investigator having to accept a statement that no differences, relationships, and/or effects will occur based on a statistical data analysis is because when nothing takes place or no differences, effects, or relationship are found there is no possible reason that can be given as to why. This is where most business managers get into trouble when attempting to offer an explanation as to why something has not happened. Attempting to provide an answer to why something has not taken place is akin to discussing how many angels can be placed on the head of a pin—everyone’s answer is plausible and possible. As such business managers need to accept that which has happened and not that which has not happened.
Many business people will skirt the null hypothesis issue by attempting to set analternative hypothesis that states differences, effects and relationships will occur between and amongst that which is being investigated if certain conditions apply.Unfortunately, however, this reverse position is as bad. The research investigator might well be safe if the data analysis detects differences, effect or relationships, but what if it does not? In that case the business manager is back to square one in attempting to explain what has not happened. Although the hypothesis situation may seem c.
Accounting for variance in machine learning benchmarksDevansh16
Accounting for Variance in Machine Learning Benchmarks
Xavier Bouthillier, Pierre Delaunay, Mirko Bronzi, Assya Trofimov, Brennan Nichyporuk, Justin Szeto, Naz Sepah, Edward Raff, Kanika Madan, Vikram Voleti, Samira Ebrahimi Kahou, Vincent Michalski, Dmitriy Serdyuk, Tal Arbel, Chris Pal, Gaël Varoquaux, Pascal Vincent
Strong empirical evidence that one machine-learning algorithm A outperforms another one B ideally calls for multiple trials optimizing the learning pipeline over sources of variation such as data sampling, data augmentation, parameter initialization, and hyperparameters choices. This is prohibitively expensive, and corners are cut to reach conclusions. We model the whole benchmarking process, revealing that variance due to data sampling, parameter initialization and hyperparameter choice impact markedly the results. We analyze the predominant comparison methods used today in the light of this variance. We show a counter-intuitive result that adding more sources of variation to an imperfect estimator approaches better the ideal estimator at a 51 times reduction in compute cost. Building on these results, we study the error rate of detecting improvements, on five different deep-learning tasks/architectures. This study leads us to propose recommendations for performance comparisons.
Explains the concept of autovalidation that can be used to select predictive models with data from designed experiments where a true validation set is not available. Contains three case studies to demonstrate the approach
Paper given at PSA 22 Symposium: Multiplicity, Data-Dredging and Error Control
MAYO ABSTRACT: I put forward a general principle for evidence: an error-prone claim C is warranted to the extent it has been subjected to, and passes, an analysis that very probably would have found evidence of flaws in C just if they are present. This probability is the severity with which C has passed the test. When a test’s error probabilities quantify the capacity of tests to probe errors in C, I argue, they can be used to assess what has been learned from the data about C. A claim can be probable or even known to be true, yet poorly probed by the data and model at hand. The severe testing account leads to a reformulation of statistical significance tests: Moving away from a binary interpretation, we test several discrepancies from any reference hypothesis and report those well or poorly warranted. A probative test will generally involve combining several subsidiary tests, deliberately designed to unearth different flaws. The approach relates to confidence interval estimation, but, like confidence distributions (CD) (Thornton), a series of different confidence levels is considered. A 95% confidence interval method, say using the mean M of a random sample to estimate the population mean μ of a Normal distribution, will cover the true, but unknown, value of μ 95% of the time in a hypothetical series of applications. However, we cannot take .95 as the probability that a particular interval estimate (a ≤ μ ≤ b) is correct—at least not without a prior probability to μ. In the severity interpretation I propose, we can nevertheless give an inferential construal post-data, while still regarding μ as fixed. For example, there is good evidence μ ≥ a (the lower estimation limit) because if μ < a, then with high probability .95 (or .975 if viewed as one-sided) we would have observed a smaller value of M than we did. Likewise for inferring μ ≤ b. To understand a method’s capability to probe flaws in the case at hand, we cannot just consider the observed data, unlike in strict Bayesian accounts. We need to consider what the method would have inferred if other data had been observed. For each point μ’ in the interval, we assess how severely the claim μ > μ’ has been probed. I apply the severity account to the problems discussed by earlier speakers in our session. The problem with multiple testing (and selective reporting) when attempting to distinguish genuine effects from noise, is not merely that it would, if regularly applied, lead to inferences that were often wrong. Rather, it renders the method incapable, or practically so, of probing the relevant mistaken inference in the case at hand. In other cases, by contrast, (e.g., DNA matching) the searching can increase the test’s probative capacity. In this way the severe testing account can explain competing intuitions about multiplicity and data-dredging, while blocking inferences based on problematic data-dredging
TEST #1Perform the following two-tailed hypothesis test, using a.docxmattinsonjanel
TEST #1
Perform the following two-tailed hypothesis test, using a .05 significance level:
· Intrinsic by Gender
· State the null and an alternate statement for the test
· Use Microsoft Excel (Data Analysis Tools) to process your data and run the appropriate test. Copy and paste the results of the output to your report in Microsoft Word.
· Identify the significance level, the test statistic, and the critical value.
· State whether you are rejecting or failing to reject the null hypothesis statement.
· Explain how the results could be used by the manager of the company.
TEST #2
Perform the following two-tailed hypothesis test, using a .05 significance level:
· Extrinsic variable by Position Type
· State the null and an alternate statement for the test
· Use Microsoft Excel (Data Analysis Tools) to process your data and run the appropriate test.
· Copy and paste the results of the output to your report in Microsoft Word.
· Identify the significance level, the test statistic, and the critical value.
· State whether you are rejecting or failing to reject the null hypothesis statement.
· Explain how the results could be used by the manager of the company.
GENERAL ANALYSIS (Research Required)
Using your textbook or other appropriate college-level resources:
· Explain when to use a t-test and when to use a z-test. Explore the differences.
· Discuss why samples are used instead of populations.
The report should be well written and should flow well with no grammatical errors. It should include proper citation in APA formatting in both the in-text and reference pages and include a title page, be double-spaced, and in Times New Roman, 12-point font. APA formatting is necessary to ensure academic honesty.
Be sure to provide references in APA format for any resource you may use to support your answers.
Making Inferences
When data are collected, various summary statistics and graphs can be used for describing data; however, learning about what the data mean is where the power of statistics starts. For example, is there really a difference between two leading cola products? Hypothesis testing is an example of making these types of inferences on data sets.
Hypothesis Tests
Claims are made all the time, such as a particular light bulb will last a certain number of hours.
Claims like this are tested with hypothesis testing. It is a straight forward procedure that consists of the following steps:
1. A claim is made.
2. A value for probability of significance is chosen.
3. Data are collected.
4. The test is performed.
5. The results are analyzed.
Hypothesis tests are performed on the mean of the population. µ
It is not possible to test the full population. For example, it would be impossible to test every light bulb. Instead, the hypothesis test is performed on a sample of the population.
Setting up a Hypothesis Test
When performing hypothesis testing, the test is setup with a null hypothesis (or claim) and the alternative hypothesis. ...
The effectiveness of various analytical formulas for
estimating R2 Shrinkage in multiple regression analysis was
investigated. Two categories of formulas were identified estimators
of the squared population multiple correlation coefficient (
2
)
and those of the squared population cross-validity coefficient
(
2 c
). The authors compeered the effectiveness of the analytical
formulas for determining R2 shrinkage, with squared population
multiple correlation coefficient and number of predictors after
finding all combination among variables, maximum correlation
was selected to computed all two categories of formulas. The
results indicated that Among the 6 analytical formulas designed to
estimate the population
2
, the performance of the (Olkin & part
formula-1 for six variable then followed by Burket formula &
Lord formula-2 among the 9 analytical formulas were found to be
most stable and satisfactory.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Data AnalysisResearch Report AssessmentBSBOllieShoresna
Data Analysis
Research Report Assessment
BSB123 Data Analysis
BSB123 Data Analysis
Notes on the Assessment
Covers Topics 1 – 10 i.e. descriptive statistics to Multiple Regression
Assignment is based around the international student recruitment industry looking specifically at students interested in postgraduate studies in USA
All 500 observations on spreadsheet are for international students
Variables are all related to factors which affect chance of being admitted and your job is to analyse this so that the company (GES) can advise future students about what to do and what their chances are of being admitted.
Report is split so that in each section you look at different aspects
You will need to do a summary incorporating elements of all of the parts to make recommendations.
Marks reflect (generally) the amount of work you need to do.
BSB123 Data Analysis
BSB123 Data Analysis
BSB123 Data Analysis
BSB123 Data Analysis
What am I looking for?
Can you select the correct technique / analysis to solve the question
Is that technique correctly and FULLY applied with calculations done correctly
E.g. in a hypothesis test, did you:
Correctly identify the test statistic (Z, T, F, χ2)
Did you include accurate hypotheses and decision rule which are consistent with each other
Were the calculations correct
Did you check to see if the assumptions or conditions of the test held
OR for Descriptive Statistics did you:
Consider all aspects of how you describe data and use the appropriate statistics to do that
Choose correct graph(s) for the type of data
Summarise the results to actually describe what you found – not just quote the stats.
Can you interpret the results – not just make a decision or complete a calculation.
Can you express the result in terms of the question and in a way which is understandable to your audience
In other words you will not get full marks unless you can correctly select the right approach to take for the data given, accurately and fully apply that analysis in a way which logically leads to a conclusion, make the conclusion in terms of the problem presented and then communicate that solution concisely and clearly
BSB123 Data Analysis
BSB123 Data Analysis
Examples from THA 4
H0: ≤ 700
H1: > 700
What is wrong with this?
BSB123 Data Analysis
BSB123 Data Analysis
Include title of analysis – t-Test: Two Sample Assuming Unequal Variances
5
Examples from THA 4
BSB123 Data Analysis
BSB123 Data Analysis
Look at t stat – all wrong – copied from somewhere – multiple students all getting it wrong
P and t test – do one
Used population terminology not sample
P-value – what is it?
6
Hypothesis Test
State the Hypotheses in terms of the parameter (µ,σ,p)
Identify the correct probability distribution (t, z, F, χ2)
Identify level of significance
State decision rule clearly
Use either test statistic method (i.e. in terms of t or z etc) or in terms of p-value. Don’t need to do both.
Decision rule must be consistent wit ...
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Review Parameters Model Building & Interpretation and Model Tunin.docxcarlstromcurtis
Review Parameters: Model Building & Interpretation and Model Tuning
1. Model Building
a. Assessments and Rationale of Various Models Employed to Predict Loan Defaults
The z-score formula model was employed by Altman (1968) while envisaging bankruptcy. The model was utilized to forecast the likelihood that an organization may fall into bankruptcy in a period of two years. In addition, the Z-score model was instrumental in predicting corporate defaults. The model makes use of various organizational income and balance sheet data to weigh the financial soundness of a firm. The Z-score involves a Linear combination of five general financial ratios which are assessed through coefficients. The author employed the statistical technique of discriminant examination of data set sourced from publically listed manufacturers. A research study by Alexander (2012) made use of symmetric binary alternative models, otherwise referred to as conditional probability models. The study sought to establish the asymmetric binary options models subject to the extreme value theory in better explicating bankruptcy.
In their research study on the likelihood of default models examining Russian banks, Anatoly et al. (2014) made use of binary alternative models in predicting the likelihood of default. The study established that preface specialist clustering or mechanical clustering enhances the prediction capacity of the models. Rajan et al. (2010) accentuated the statistical default models as well as inducements. They postulated that purely numerical models disregard the concept that an alteration in the inducements of agents who produce the data may alter the very nature of data. The study attempted to appraise statistical models that unpretentiously pool resources on historical figures devoid of modeling the behavior of driving forces that generates these data. Goodhart (2011) sought to assess the likelihood of small businesses to default on loans. Making use of data on business loan assortment, the study established the particular lender, loan, and borrower characteristics as well as modifications in the economic environments that lead to a rise in the probability of default. The results of the study form the basis for the scoring model. Focusing on modeling default possibility, Singhee & Rutenbar (2010) found the risk as the uncertainty revolving around an enterprise’s capacity to service its obligations and debts.
Using the logistic model to forecast the probability of bank loan defaults, Adam et al. (2012) employed a data set with demographic information on borrowers. The authors attempted to establish the risk factors linked to borrowers are attributable to default. The identified risk factors included marital status, gender, occupation, age, and loan duration. Cababrese (2012) employed three accepted data mining algorithms, naïve Bayesian classifiers, artificial neural network decision trees coupled with a logical regression model to formulate a prediction m ...
1. 1
Theoretical Formula Scoring Criterion to Simulate
Correction for Guessing
Abstract
One of the major phenomenon's in standardized multiple-choice tests is determining how much
random guessing occurs at a particular question. It can be intuitive to think that when a multiple-choice
test is admitted the probability of an individual guessing an item correct is one out of all possible answer
choices if they truly do not know the correct answer. Even with this kind of attempt, guessing potentially
increases error in measuring an individual's true score, let alone their true ability. A measurement
framework, known to many as formula-scoring, was proposed several years ago to possibly help the
correction for guessing. The purpose of this proposal is to use the general framework of the formula-
scoring method and introduce a new formula-scoring criterion that can be implemented through
simulation. This new criterion will show that if formula-scoring is applied then we may be able to
improve an individual's overall observed score (sum score) to reflect what their true score really is.
.
2. 2
I. Purpose
The idea of formula scoring is that a multiple-choice test being admitted should provide detailed
instructions as to how a test-taker should approach attempting an item. The rule generally follows that an
examinee should omit an item if they experience a situation of having to guess (whether they have partial
or no knowledge of the item). If they attempt an item and get it correct, they will receive a score of 1 (this
is based on if they have more than partial knowledge of the item). If they experience a situation where
they must rely on some type of guessing they should, per the instructions, omit the item and receive a
score of 0. A consequence of answering an item incorrectly is receiving a score of -1/(k-1) where k is the
number of possible answers to a multiple-choice item. When these rules are in place the complete formula
score for potentially correcting error in guessing is,
where S is the individual's corrected score, R is the number of items the individual marked correctly, W is
the number of items the individual marked incorrectly and k is the number of possible choices for each
item (Diamonds and Evans, p. 181). If guidelines such as the one listed above are given to individuals
taking the test, it's possible there may be improvements obtaining true scores through the individuals
observed score. It would seem that just implementing this in all tests would resolve the issue of guessing,
but the effects of guessing behaviors that occur can vary greatly. Such acts of guessing can cause the test-
taker to think about the possible rewards of a higher score if they attempt all items. Some of these
guessing behaviors are, but not limited to:
" 1.Eliminating one or more answer choices judged to be definitely wrong
2.Making use of unintended semantic or syntactic cues available from the wording of the questions or
the response options
3.Falling into traps set by the ingenious item writer e.g. cliché-like choices that sound plausible but are
wrong
4.Responding on the basis of some element in one of the response choices that attracts him, but at a
relatively low level of confidence
5.Using some essentially random fashion of responding-in the extreme, flipping a coin or marking some
specific response position or pattern of positions " (Thorndike, p. 59)
Many testing services implement the formula-scoring rule mentioned previously. Testing
companies such as ETS (Educational Testing Services) have employed formula scoring on the SAT
(Scholastic Aptitude Test) and the GRE (Graduate Record Examination) subject exams since 1953
(Budescu and Bar-Hillel, p. 278). However some researchers have suggested that the use of this formula
score correction can reduce the reliability of a test. In contrast to these claims, studies have shown that the
reliability between corrected and uncorrected scores, if guessing instructions were implemented, were not
that far off from each other (Diamonds and Evans, p. 182).With that being said, the purpose of this paper
is to propose a formula-scoring criterion (FSC) that can be implemented through simulations that may
show signs of an individual's observed score falling closer to their true score.
3. 3
II. Theoretical Framework
In standardized multiple-choice tests such as a Math or English placement exam, a person's
ability can best be measured using unidimensional Item Response Theory (IRT) models. What is obtained
from these models is the probability of an examinee getting one individual item correct; assuming we
know their true ability, how difficult the problem is, how discriminating the item is and how much
guessing is potentially involved with this particular item. This model (known as a three-parameter
logistics or 3PL model) is typically written as,
where Uij is item i attempted by examinee j, θj is the examinees given ability, ai is the discrimination value
on the ith
item, bi is the difficulty value on the ith
itemand ci is the guessing value on the ith
item. In order
for this model to potentially hold true, three assumptions must be met: 1) θ is unidimensional, 2) Local
Independence, and 3) Monotonicity. Another alternative IRT model that will be presented in this paper is
a 3PL model created by Peter J. Pashley. Pashley elegantly designed a 3PL model around what he calls a
four-parameter hyperbolic equation. The purpose of incorporating this model is to explore other models,
not widely used in IRT, to compare the proposed FSC. Pashley proposed the following model parameters,
where f defines a slope parameter, h defines the difficulty parameter, and k is thought of as the lower
asymptote parameter. With these new parameters, Pashley formulized the four-parameter hyperbolic
model as such,
(Pashley, pgs. 9-10).
In many IRT studies, simulations are carried out where computer software, such as R, can
simulate random examinees taking a multiple-choice item test. What is produced through these
simulations are datasets consisting of a very large matrix of 1's (right) and 0's (wrong) where each row
represents a test-taker and each column represents score patterns for each multiple-choice item. One
problem with these simulations is they are not able to take into consideration a multiple-choice test
applying the formula-scoring rule. This paper proposes an FSC, let it be defined as τ, that can be used in
the practice of simulating IRT data coming from multiple-choice tests being implemented with formula-
score rules. The FSC was developed around the assumption of correlations between all estimated c
4. 4
parameters and various subgroups of estimated θ's. What was discovered was a vector (denoted as ) of
these correlations fell under a pattern very similar to a normal distribution. That is,
where j = the number of items on the test and l = the number of simulated examinees. With this in place, τ
will be defined as a random number from this hypothesized normal distribution. Therefore, the following
can be applied to a simulated examination taken by random examinees with formula-scoring applied,
III. Methods & Data
To defend the FSC, two separate simulations were undertaken in R: 1) A simulation of all
parameters assumed to be known and 2) a parametric bootstrap sampling procedure. In order to carry out
an appropriate simulation, real data was used which consisted of 2,642 examinees attempting a 32
multiple-choice item Form B College Math Placement test. Parameters were estimated from this data
through BILOG-MG using Expected A Posteriori (EAP), a type of Bayesian approximation method. The
following parameters estimated through BILOG-MG followed the distributions given in Table 1.
Table 1. Parameter Distributions
Parameter Distribution
a LogNormal(0.290,0.301)
b Normal(0.246,0.962)
c Beta(2.771,14.021)
IV. Results
In the first simulation, 30,000 simulated examinees took a 32-item test three times. Each test
followed these categories: a standard test (0 wrong and 1 correct), a test implementing the FSC, and a test
implementing the FSC using Pashley's 3PL model. In Table 2 (on p. 5), a sample of 10 examinees were
presented with their observed score and true score (calculated under methods of Classical Test Theory).
As you can see in the table, it's surprising how some examinee's observed score comes fairly close to the
their true score when taking the formula-scoring test. Based on this FSC, there were however instances
where simulated examinees scored well under or above their true score (examinee 7 for example). In the
second simulation, a parametric bootstrap procedure was conducted with the same number of examinees
and items as compared to the real data. From Table 3 (on p. 5), it's noticed that the reliability (Guttman's
λ2) of the simulated tests (applying the FSC and/or Pashley's model) were less than the reliability of the
5. 5
simulated standard test. This would make sense being that Ruch and Stoddard claimed this to be true back
in 1925 (Diamonds and Evans, p. 182). However, notice how the values of the standard deviation of the
biserials show a reduction in the FSC method and Pashley's model applied with the FSC method.
However a negative aspect were the results of the SEM. It seemed the test's SEM increased with the FSC
method and with Pashley's model with the FSC method. Although SEM did increase, it wasn't by much.
In conclusion, the next 3-5 months will be spent analyzing this new FSC method to ensure its validity and
statistical power in simulating an examination undertaken with formula-scoring rules.
Table 2. Simulated Examinees
Observed Score
Examinee True Ability Standard
Formula-
Scoring
Pashley w/
Formula-
Scoring
True Score True SD
1 0.4564 24.0000 19.2500 17.2500 20.9410 2.2180
2 -0.0309 17.0000 18.0000 19.7500 16.5580 2.3520
3 0.2706 22.0000 18.7500 17.0000 19.3030 2.2960
4 -0.5884 11.0000 11.7500 8.2500 12.2310 2.3330
5 -1.1745 11.0000 5.7500 9.7500 8.7980 2.2710
6 0.1921 18.0000 17.0000 13.7500 18.5840 2.3210
7 -0.5151 14.0000 9.7500 16.7500 12.7380 2.3380
8 -0.3514 17.0000 15.2500 12.5000 13.9340 2.3470
9 1.7278 27.0000 30.0000 29.7500 28.7620 1.4800
10 0.2953 19.0000 19.5000 17.7500 19.5280 2.2870
Table 3. Parametric Bootstrap Simulation (500 Iterations)
Statistics Real Data Standard 3PL
Standard 3PL w/
Formula-Scoring
Pashley's 3PL w/
Formula-Scoring
Guttman's λ2 0.8396 0.8976 0.8940 0.8883
SEM 2.4759 2.3190 2.3491 2.3727
sd(ρbis) 0.1003 0.1412 0.1358 0.1211
V. Educational Significance
When it comes to issuing standardized multiple-choice tests, such as a Math/English college
placement test, various acts of guessing can occur that can result in inaccurate IRT measurements of a
person's true score, which could potentially yield under/over estimates of their true ability. Based on these
simulations using the proposed FSC criterion, implementing this type of formula-scoring rule to multiple-
choice testing practices in the future may open doors in improving a person's overall observed score being
very close to their overall true score. Although there are flaws in the suggested method, optimistically its
logical reasoning will be accepted.
6. 6
References
Bar-Hillel, David Budescu and Maya. "To Guess or Not to Guess: A Decision-Theoretic View of Formula
Scoring." Journal of Educational Measurment, Vol. 30, No. 4 (Winter, 1993): 277-291.
Evans, James Diamond and William. "The Correction For Guessing." Review of Educational Research,
Vol. 43, No. 2 (Spring, 1973): 181-191.
Habing, Dr. Brian. STAT 778 - Spring 2014 - R Templates. 2014. 2014
<http://www.stat.sc.edu/~habing/courses/778rS14.html>.
Michele Zimowski, Eiki Muraki, Robert Mislevy, and Darrell Bock. BILOG-MG 3. Scientific Software
International, Inc. Stokie, IL, 2005-2014.
Pashley, Peter J. An Alternative Three-Parameter Logistic Item Response Model. Research. Princeton,
New Jersey: Educational Testing Service, February 1991.
Thorndike, Robert L. Educational Measurement: The Problem of Guessing. Washington, D.C.: American
Council on Education, 1971.