18 Asian Journal of Education and Training Vol. 4, No. 1, 18-28, 2018 ISSN(E) 2519-5387 DOI: 10.20448/journal.522.2018.41.18.28 Test of Achievement in Quantitative Economics for Secondary Schools: Construction and Validation Using Item Response Theory Lydia I. Eleje1  Nkechi P.M. Esomonu2 ( Corresponding Author) 1,2Department of Educational Foundations, Faculty of Education, Nnamdi Azikiwe University, Awka, Nigeria Abstract A Test to measure achievement in quantitative economics among secondary school students was developed and validated in this study. The test is made up 20 multiple choice test items constructed based on quantitative economics sub-skills. Six research questions guided the study. Preliminary validation was done by two experienced teachers in secondary school economics and one expert in test construction. The pilot testing was conducted to ensure compatibility of the test items while test try-out on 515 students yielded the data for item analysis. Analysis were attained through SPSS and Bilog MG using Item Response Theory (IRT) three-parameter logistic model (3PL) to establish item difficulty, item discrimination, and the guessing value. The empirical reliability of the test was 0.86. The test was found to be of good quality, valid and highly reliable. The test is ready and can now be used to assess students’ achievement in quantitative economics in Nigeria and other countries. Keywords: Achievement test, Quantitative economics, IRT (Item Response Theory), Uni-dimensionality, Reliability. Citation | Lydia I. Eleje; Nkechi P.M. Esomonu (2018). Test of Achievement in Quantitative Economics for Secondary Schools: Construction and Validation Using Item Response Theory. Asian Journal of Education and Training, 4(1): 18-28. History: Received: 4 December 2017 Revised: 11 January 2018 Accepted: 28 January 2018 Published: 25 January 2018 Licensed: This work is licensed under a Creative Commons Attribution 3.0 License Publisher: Asian Online Journal Publishing Group Contribution/Acknowledgement: Both authors contributed to the conception and design of the study. Funding: This study received no specific financial support. Competing Interests: The authors declare that they have no conflict of interests. Transparency: The authors confirm that the manuscript is an honest, accurate, and transparent account of the study was reported; that no vital features of the study have been omitted; and that any discrepancies from the study as planned have been explained. Ethical: This study follows all ethical practices during writing. Contents 1. Introduction ...................................................................................................................................................................................... 19 2. Literature Review ............................................................................................................................................................................ 19 3. Method ............................................................................................................................................................................................... 21 4. Results ................................................................................................................................................................................................ 23 5. Discussion of Findings .................................................................................................................................................................... 24 6. Conclusions ....................................................................................................................................................................................... 24 7. Recommendations ............................................................................................................................................................................ 24 References .............................................................................................................................................................................................. 24 http://crossmark.crossref.org/dialog/?doi=10.20448/journal.522.2018.41.18.28&domain=pdf&date_stamp=2017-01-14 http://creativecommons.org/licenses/by/3.0/ http://creativecommons.org/licenses/by/3.0/ https://orcid.org/0000-0002-8587-289X https/orcid.org/0000-0002-8682-382x https://orcid.org/0000-0002-8587-289X https/orcid.org/0000-0002-8682-382x https://orcid.org/0000-0002-8587-289X https/orcid.org/0000-0002-8682-382x Asian Journal of Education and Training, 2018, 4(1): 18-28 19 1. Introduction The teaching and learning of economics as a subject in Nigeria and other West African countries is part of the senior secondary- SS1, SS2 and SS3 curriculum (Federal Government of Nigeria, 2004). Economics according to Kallie (2015) is a social science for analyzing the production, distribution, and consumption of goods and services. Economics education is the education provided for economics teachers or would be teachers on different economic issues and problems. It focuses on the improvement of economic curriculum materials and pedagogy. This according to Babalola (2003) is the education provided to assist a rational man in organizing his different thoughts whenever he is faced with day-to-day economic issues and problems. Thus, economics education is aimed at bringing about desirable behavioral changes in students (Dike, 2002). These behavioral changes are quantified and qualified using achievement tests. Achievement tests are tests given to assess how far a student has learnt what was taught. They are primarily used in making classroom level decisions and are designed with particular reference to the course objectives/learning goals of a specific course, study program or class (Mahajan, 2015). Such tests evaluate students’ understanding of a particular instructional domain in order to make decisions regarding the advancement or knowledge and skill acquired by the students. Achievement test can be influenced by certain factors. This is supported in a research by Adhanja et al. (2016) and Hanzen (2000) which shows that students’ achievement is affected by different factors such as learning abilities, race, gender, sex and socio-economic factors. On the other hand, Chansarkar and Michaeloudis (2001) explained that students’ achievement is not affected by such factors as age, sex and place of residence but is associated with performance in quantitative skills/subject. They also found out that those who live near school perform better than other students. In support of Chansarkar and Mishaeloudis view, Educational Testing Service (1995) also identified quantitative ability/skill as a factor which could influence students’ achievement in a school subject. Dolado and Morales (2008) specifically pointed out ability in calculations/mathematics as a factor affecting performance in economics. Quantitative ability is seen as a measure of a student’s ability/skill to apply knowledge of mathematical concepts and principles, to demonstrate flexibility in thinking, to identify critical features on new situations, to make correct generalizations, and to compare mathematical expressions (ETS, 1995; Dolado and Morales, 2008). Quantitative skills involve the ability to solve numerical problems easily. To some people, quantitative ability also means that they are able to separate a whole into its constituent parts more readily than others. It includes arriving at mathematical/calculative solutions to problems, as well as basic adding, subtracting, multiplying and dividing (Adu et al., 2009). Since the introduction of a new economics syllabus which incorporated some elements of quantitative/calculations into the subject, students’ achievement in economics has not been as good as it has been before. This new syllabus has been a challenge for the economics students in the senior secondary classes partly as a result of the carry over effects of the negative attitudes which they have towards quantitative/calculations (Adu et al., 2009). Consequently, the comments made by West Africa Examination Council (WAEC) Chief examiners tend to portray the impact of quantitative ability on students’ achievement in economics. In West African Examination Council (WAEC) (2010) and WAEC (2012) the chief examiners’ in their reports specifically categorized students’ weaknesses as: i. Calculations- majority of the students/candidates were unable to carry out simple calculations. A question in price elasticity of demand was attempted poorly by very few candidates. ii. Poor graph/curve analysis- candidates attempted unsatisfactorily to draw and analyze graphs/curves. The Chief examiners’ also suggested that students should be encouraged to develop their quantitative skills/ability in economics before the examinations. The need for students to develop their quantitative skills in economics was also emphasized by Osadebe (2014). Osadebe pointed out that a good knowledge in quantitative economics at senior secondary will later help students in tertiary schools to study Economics, Banking, Finance, Accounting and other related courses involving calculations. Hence, there is need for a quality tests of achievement in quantitative economics to enable senior secondary economics teachers’ asses their students’ level of performance/achievement in quantitative economics prior to external examinations. To ensure the quality of the test of achievement in quantitative economics being developed, item analysis of students’ responses to the test items were carried out using Item Response Theory (IRT). Although Eleje et al. (2016) developed and validated diagnostic economics test on three content areas and Esomonu and Eleje (2017) also developed and validated diagnostic quantitative economics skill test using IRT, these tests were diagnostic tests not achievement test. So, there is lack of test of achievement in quantitative economics for secondary schools to the best of our knowledge. In essence, the construction and validation of test of achievement in quantitative economics has not been researched on, thereby constituting an educational need. The researchers hence deem it necessary to construct and validate test of achievement in quantitative economics for secondary schools using item response theory (IRT). The objective of this study therefore, is to construct and validate test of achievement in quantitative economics (TAQE) for secondary schools using IRT. Based on the objective stated above, the study sought answers to the following research questions: 1. Is the test unidimensional? 2. Which of the IRT model represents a better fit for the test data? 3. What are the test item threshold levels? 4. What are the test item slope values? 5. What are the test item guessing values? 6. How reliable is the entire test according to IRT model? 2. Literature Review The literature was reviewed based on the following sub-headings: Item response theory (IRT) The a parameter Asian Journal of Education and Training, 2018, 4(1): 18-28 20 The b parameter The c parameter Quantitative economics skill Validity of the test Reliability of the test 2.1. Item Response Theory (IRT) Thorpe and Favia (2012) describe item response theory (IRT) as a collection of measurement models that try to explain the connection between observed item responses on a scale and an underlying construct. IRT models are mathematical equations that specifically describe the association between subjects’ levels on a latent variable and the probability of a particular response to an item, using a non-linear monotonic function (Hays et al., 2009). As a measurement theory IRT aids to develop an educational measurement scale that has ratio scale, sample independent attributes and students’ ability reported on both item and total instrument levels (Ani, 2014). IRT makes it possible to estimate and interpret item statistics referred to as parameters. In IRT, parameters of the persons are invariant across items, and parameters of the items are invariant in different populations of persons. Thus it brings greater flexibility and provides more sophisticated information which allows for the improvement of the reliability of an assessment (Ani, 2014). Item parameters in IRT are estimated directly using three logistic models instead of proportions (difficulty or threshold) and item-scale correlations (discrimination) (Adedoyin and Mokobi, 2013). In one-parameter logistic (1PL) model, items are only described by a single parameter in terms of location or difficulty (b) while the slopes (discrimination) are held constant. The two-parameter logistic (2PL) model not only checks for the item difficulty but also assess each item’s level of discrimination between high and low ability students while the three-parameter logistic (3PL) model adds a third item parameter which is called pseudo-guessing parameter that reflects the probability that an examinee with a very low trait level will correctly answer an item solely by guessing. This according to Ani (2014) implies that students can through guessing answer an item correctly in an achievement test. 2.2. The A Parameter The a parameter is a measure that can be graphically expressed by the steepness of the item characteristic curve (ICC). The a parameter also known as item discrimination (slope) shows how well an item can differentiates among respondents with abilities to the left of the item location from those with abilities to the right of the item location (Thorpe and Favia, 2012). If a parameter value is positive, it implies that students’ of higher ability have a high probability of answering an item correctly and students’ of lower ability have a low probability to answer an item correctly. When a parameter value is negative, it indicates that students’ with high ability have a low probability of answering an item correctly while students’ with low ability have a higher probability of answering an item correctly. A high discrimination value indicates that the item discriminates well among low and high skilled students. The discrimination values (parameter a-values) of good items according to Adedoyin and Mokobi (2013) ranges from 0.5 to 2. If the values of the item discrimination a is above 1, this is normally desirable value for a good test item. Baker (2001) describes the range and interpretation of values for item discrimination as follows: very low; .01 - .34, Low; .35 - .64, moderate; .65 - 1.34 High; 1.35 - 1.69 and Very high; 1.70 and above. 2.3. The B Parameter The b parameter also known as item difficulty (threshold), is a location index along the x-axis that tells us how easy or how difficult an item is. The index of an item’s location is the point on the x-axis at which the curve crosses the 0.5 probability value on the y- axis. Negative difficulty estimates indicate that the items are easy while positive difficulty estimates indicate that the items are hard. An easy item functions among the low-ability examinees; a difficult item functions among the high-ability examinees (Baker, 2001; Thorpe and Favia, 2012). The values of b greater than 1 indicate a very difficult item and items with low b values below -1 indicate easy items. When the values of b are between -0.5 and 0.5, then the test items with such difficulty indexes have medium difficulty level. Baker (2001) pointed out that theoretically, difficulty values can range from - 00 to + 00, in practice, difficulty values usually are in the range of - 3 to + 3. Ceniza and Cereno (2012) gave the interpretation for values of item threshold/difficulty (b-values) as follows: Very Easy = Less than -2, Easy = -0.50 to -2.00, Average = -0.49 to 0.49, Difficult = 0.50 to 2.00 and Very Difficult = Greater than 2.00. 2.4. The C Parameter The c parameter also known as a pseudo-guessing parameter (Thorpe and Favia, 2012) is the likelihood that an examinee with very low ability can guess the correct response to an item and therefore has a greater-than-zero probability of answering correctly. An examinee who randomly selects responses to items of four options can answer these items correctly about 1 out of 4 times, meaning that the probability of guessing correctly is about 0.25. Harris (1989) concluded that the items with 0.30 or greater c-values are considered not very good, rather c- values of 0.20 or lower are desirable. In like manner, Akindele (2003) also noted that items do not have perfect c- values because examinees do not guess randomly when they do not know the answer. 2.5. Quantitative Economics Skills Quantitative skills involves the ability to solve numerical problems easily. It entails being able to separate a whole into its constituent parts more readily than others. It includes arriving at mathematical/calculative solutions to problems, as well as basic adding, subtracting, multiplying and dividing (Adu et al., 2009; Riley, 2015). The quantitative skills in secondary economics as listed by Riley (2015) are as follows: A) Calculate, use and understand ratios and fractions B) Calculate, use and understand percentages and percentage changes Asian Journal of Education and Training, 2018, 4(1): 18-28 21 C) Understand and use the terms mean, median and relevant quartiles D) Construct and interpret a range of standard graphical forms E) Calculate and interpret index numbers F) Calculate cost, revenue and profit (marginal, average, totals) G) Make calculations to convert from money to real terms H) Make calculations of elasticity and interpret the result I) Interpret, apply and analyze information in written, graphical and numerical forms, Riley (2015). 2.6. Validity of the Test The validity of the test developed using IRT is dependent on the extent to which the students’ responses (data) fit the model or if the model is appropriate for the data. Goodness of fit statistics can be used to test for the model fit to the data. But, -2Log likelihood value is commonly used in Bilog MG to check the goodness of model fit, where higher values indicate a poorer fit of the data to the model. Comparing the values from different models can indicate which model represents a better fit (Thorpe and Favia, 2012; Rijn et al., 2016). The analysis of model fit is a check on internal validity (Obinne, 2013). According to Korashy (1995) if the model fit is acceptable, then the item is valid. However, the smallest -2Log likelihood value is the best (Thorpe and Favia, 2012). Three parameter logistic model was used in this study because it has the lowest -2Log likelihood value. 2.7. Reliability of the Test Reliability refers to the consistency of a measuring instrument. A test is considered reliable if the same result is approximately gotten repeatedly (Cherry, 2009). Unlike classical test theory that requires another test for reliability, IRT has a local reliability. That is, an amount of information at each point of underlying continuum (Ceniza and Cereno, 2012). With IRT model, each item of the test contained information. For each parameter logistic model, Bilog MG computed an empirical reliability index. A reliability index within the range of 0.81to 1.0 indicates high reliability; 0.61 to 0.80 shows a moderate reliability; 0.41 to 0.60 means fair reliability; 0.10 to 0.40 means slight reliability and less than 0.10 means virtually no reliability. 3. Method The study is an instrumentation research. It is an instrumentation research since the objective of the study is to construct, validate and produce a valid and reliable test assessing students’ achievement in quantitative economics. The researchers constructed and validated test of achievement in quantitative economics for secondary schools by following these test development stages which are discussed thereafter: • Planning the test • Constructing the test • Initial validation of the test • Pilot testing • The test try-out • Item Analysis • Selection of good items 3.1. Planning the Test Planning the test involves identification of the study population, sample and sampling technique, and the test content. A total of 515 students were randomly selected from all the senior secondary three (SS3) economics students in Anambra State in south-east Nigeria as the test try-out sample. The test content area is based on the quantitative sub-skills in secondary economics as given by Riley (2015). Table 2 shows various units of the quantitative economics sub-skills listed along the rows while different educational objectives to be tested were listed along the columns. Table-1. Economics Achievement Test Table of Specification Content Know Com Appli Interpr Total Calculate, Use and Understand Ratios and Fractions - 1 2 - 3 Calculate, Use and Understand % and % Changes 1 - 3 - 4 Understand and Use Mean, Median and Relevant Quartiles 2 - 1 - 3 Construct & Interpret a Range of Standard Graphical Forms 1 2 - 2 5 Calculate and Interpret Index Numbers 1 - - 1 2 Calculate Cost, Revenue & Profit (Marginal, Average & Totals) 1 - 1 - 2 Make Calculations to Convert Money to real Terms - - 2 1 3 Make Calculations of Elastitcity & Interpret the Result 2 2 1 1 6 Interpret, Apply and Analyze Numerical Forms 1 - 1 - 2 Total 9 6 11 5 30 Abbreviations: Know=Knowledge; Com=Comprehension; Appli=Application; Interpr=Interpretation. 3.2. Constructing the Test The researchers constructed 30 multiple choice format objective test items that is in line with the table of specifications using economics text books recommended by the Ministry of Education (Anyaele, 2003; Anyanwuocha, 2006). The items of test of achievement in quantitative economics (TAQE) constructed were based on the secondary economics quantitative sub-skills as stipulated by Riley (2015) (see Table 1) and guidelines given by Alderson (2005) and Winarni (2002). Asian Journal of Education and Training, 2018, 4(1): 18-28 22 3.3. Initial Validation of the Test In its initial stage, the 30 items constructed were face and content validated by one expert in educational measurement, evaluation and research, and two experienced secondary economics teachers. Their expert observations, comments and suggestions were used in the modifications of the test. 3.4. Pilot Testing In its preliminary stage, the 30 items constructed were pilot tested on few SS3 economics students to check grammatical error and compatibility of test of achievement in quantitative economics skill items. 3.5. The Test Try-Out The test was tried out on 515 SS3 economics students. The test try-out yielded the data for item analysis. 3.6. Item Analysis The students’ responses to the 30 items of the test of achievement in quantitative economics (TAQE) were subjected to item analysis using Item Response Theory (IRT) to ensure the quality of the items. The items of the test were first checked to determine the dominance of the first factor (uni-dimensionality), secondly assessed for model fit, before IRT psychometric analysis were done to estimate the test items parameters (a, b, c-values) using data from students’ responses. Test for uni-dimensionality in this study was exploratory factor analysis. It is to find out if there exist a dominant trait underlining achievement in quantitative economics assessed by the TAQE. Test for model fit was also assessed by comparing the values of -2Log likelihood from different models to indicate which model represents a better fit for the data. The thirty (30) items of test of achievement in quantitative economics were subjected to IRT psychometric analysis using Bilog MG software to estimate item threshold, slope and asymptote (difficulty, discrimination and guessing) values. 3.7. Selection of Good Items Items retained in the final output of the test were based on the verbal interpretation of each item as illustrated by Ceniza and Cereno (2012) in Table 2 below. The 20 items that was included in the final output of the TAQE were arranged in-line with the quantitative sub-skills in secondary economics. Table-2. Decision table of difficulty and discrimination indices Difficulty Level Discrimination Level Decision Easy Low Moderate High Revise Retain Retain Average Low Moderate High Revise Retain Retain Difficult Low Moderate High Revise Retain Retain Source: Ceniza and Cereno (2012) Table-3. Total Variance Explained by the result of factor analysis Initial Eigenvalues Component Total % of Variance Cumulative % 1 7.407 24.691 24.691 2 2.359 7.863 32.554 3 1.648 5.494 38.048 4 1.372 4.573 42.620 5 1.240 4.132 46.752 6 1.130 3.767 50.519 7 1.077 3.589 54.108 8 .997 3.323 57.430 9 .933 3.109 60.540 10 .864 2.881 63.420 11 .814 2.713 66.133 12 .768 2.560 68.693 13 .756 2.520 71.213 14 .731 2.437 73.650 15 .701 2.337 75.987 16 .628 2.093 78.080 17 .607 2.024 80.103 18 .583 1.942 82.045 19 .558 1.861 83.906 20 .530 1.765 85.671 21 .517 1.724 87.395 22 .498 1.659 89.054 23 .487 1.622 90.677 24 .473 1.576 92.253 25 .458 1.526 93.779 26 .412 1.372 95.151 27 .387 1.289 96.440 28 .379 1.263 97.704 29 .368 1.227 98.931 30 .321 1.069 100.000 Extraction Method: Principal Component Analysis. Asian Journal of Education and Training, 2018, 4(1): 18-28 23 4. Results 4.1. Research Question 1 Is the test uni-dimensional? Table 3 is the result of exploratory factor analysis done on test of achievement in quantitative economics (TAQE) for uni-dimensionality assessment. The result shows that TAQE had six eigen values greater than one and the remaining twenty four values less than one. The first eigen value is 7.407, followed by 2.359, 1.648, 1.372, 1.240, etc. 4.2. Research Question 2 Which of the IRT model represents a better fit for the TAQE data? Table-4. Model fit information -2Log likelihood1-PL = 17773.820 -2Log likelihood2-PL = 17370.596 -2Log likelihood3-PL = 17140.261 Source: results from Bilog MG analysis Three (3) parameter logistic model in Table 4 had the lowest -2Log likelihood. 4.3. Research Questions 3, 4 and 5 What are the item threshold, slope and guessing values/levels of DQEST items based on three parameter logistic (3PL) model? Table-5. Item threshold (difficulty estimates), slope (discrimination), and asymptote (guessing) values of DQEST items. Item Threshold (Difficulty) Difficulty Level Slope (Discrimination) Discrimination Level Asymptote (Guessing) Remark 1 0.16 Average 0.21 Low 0.08 Revise 2 0.13 Average 0.16 Low 0.06 Reject 3 0.12 Average 0.20 Low 0.06 Revise 4 0.10 Average 0.20 Low 0.05 Revise 5 0.11 Average 0.20 Low 0.05 Revise 6 0.12 Average 2.47 Very High 0.02 Reject 7 0.06 Average 2.29 Very High 0.02 Reject 8 0.47 Average 0.08 Low 0.08 Reject 9 0.38 Average 0.07 Low 0.07 Reject 10 0.61 Difficult 0.05 Low 0.09 Reject 11 0.51 Difficult 0.09 Low 0.07 Reject 12 0.08 Average 1.15 Moderate 0.03 Retain 13 0.08 Average 1.53 High 0.04 Retain 14 0.08 Average 0.24 Low 0.05 Revise 15 0.12 Average 2.21 Very High 0.01 Reject 16 0.06 Average 1.20 Moderate 0.02 Retain 17 0.07 Average 0.35 Moderate 0.04 Retain 18 0.06 Average 1.08 Moderate 0.02 Retain 19 0.06 Average 2.17 Very High 0.02 Revise 20 0.10 Average 0.24 Low 0.05 Revise 21 0.09 Average 0.31 Low 0.04 Revise 22 0.09 Average 0.25 Low 0.04 Revise 23 0.08 Average 2.73 Very High 0.02 Reject 24 0.10 Average 2.55 Very High 0.02 Reject 25 0.07 Average 0.45 Moderate 0.03 Retain 26 0.11 Average 0.46 Moderate 0.04 Retain 27 0.07 Average 1.14 Moderate 0.03 Retain 28 0.09 Average 0.37 Moderate 0.03 Retain 29 0.09 Average 0.91 Moderate 0.03 Retain 30 0.06 Average 1.57 High 0.02 Retain Source: results from DQEST item analysis based on 3pl model As Table 5 shows, the item threshold (difficulty) and item slope (discrimination) values indicated that eleven (11) items be retained while nine (9) items (1, 3, 4, 5, 14, 19, 20, 21 and 22) be revised. The range of the guessing value were within 0.00 to 0.09. 4.4. Research Question 6 How reliable is the entire test according to IRT model? Table-6. Summary statistics from item analysis results N 515 Mean -0.009 Standard Deviation 1.012 Variance 0.1407 Empirical Reliability 0.8629 Source: results from Bilog MG analysis Asian Journal of Education and Training, 2018, 4(1): 18-28 24 The empirical reliability index as given by Bilog MG in the three parameter logistic model is 0.86. 5. Discussion of Findings The discussion of findings from this study was based on the results gotten from assessment of dimensionality, model fit, item threshold (difficulty), slope (discrimination), guessing, and reliability values of test of achievement in quantitative economics (TAQE). Table 3 result is the exploratory factor analysis done to assess the test’s dimensionality. The findings from the Table 3 shows that the first seven (7.407, 2.359, 1.648, 1.372, 1.240, 1.130 and 1.077) eigen values were greater than one and the first eigen value which was 7.407 explained 24.691% of the variance in the data set. The rest of the variance was explained by the other 29 eigen values. This is an indication that a dominant component or factor referred to as the ability measured by the test exist among the items (Hambleton et al., 1991). Thus, test of achievement in quantitative economics developed had quantitative economics as a dominant factor. This is supported by Rijn et al. (2016). They pointed out that violation of dimensionality assumption would result in inadequacy of the model in describing the data and hence unreliable estimation of the examinee’s ability and the correct specification of the number of the latent dimensions is directly tied to the construct validity of the test. As seen in Table 4, -2Log likelihood value for IRT one parameter (1PL), two parameter (2PL) and three parameter (3PL) logistic models were 17773.820, 17370.596 and 17140.261 respectively. 3PL with the lowest - 2Log likelihood value represents the model with a better fit for the test items. The essence of using the best model fit for the items according to Thorpe and Favia (2012) is to ensure the validity of the test items. Hence, three parameter logistic (3PL) model was used in this study to estimate the item parameters. Also, the values of TAQE IRT analysis of item parameters as seen in Table 5 revealed that eleven (11) items be retained, nine (9) items be revised and ten (10) items be rejected. The selection of 11 items retained based on the b- value range of -2 to +2 and a-value range of 0.01 - .34 as Low, 0.35 - 1.34 as moderate, 1.35 - 2.00 as High, 2.01 and above as Very high was in agreement with the criteria description of Baker (2001). The 9 items numbering 1, 3, 4, 5, 14, 19, 20, 21, 22, were modified or revised and then added to the final output of the test. The twenty (20) items that made up the final output of TAQE had a desirable guessing value, that is, the probability of getting the answer correctly by mere guessing was low. The findings of this study are in line with that of Ani (2014) and Adedoyin and Mokobi (2013). The empirical reliability index of TAQE as seen Table 6 is 0.8629 (approximately 0.86). This indicates 86% certainty of the consistency of the test items in yielding approximately same result repeatedly and that the reliability of test of achievement in quantitative economics (TAQE) developed was high. Thus, the output of this study will provide a valuable instrument in the field of economics education in Nigeria and other countries. 6. Conclusions The growing interest in the use of IRT is an evidence of the growth in psychometrics, and computer adaptive testing in particular (Embretson and Reise, 2000). According to Hays et al. (2000) IRT has a number of potential advantages over CTT in assessing learning, in developing better measures and in assessing change over time. Its models yield invariant item and latent trait estimates. IRT psychometric methodologies have been used to solve assessment challenges as identified by Aiken (2003) and Cook et al. (2003). Since educational tests are the main source of information about student achievement in schools and in the context of large-scale testing the analysis of test data is essential in determining the quality of the test and the information the test generates. The worth of any educational assessment endeavor depends on the instruments i.e. the tools and techniques used, if these instruments are poorly designed, the assessment can be a waste of time and money (Adedoyin and Mokobi, 2013). Based on the findings of this study, in terms of results from dimensionality assessment, model fit assessment, values from IRT analysis of Item threshold, slope, guessing, and reliability, the researchers therefore conclude that the test of achievement in quantitative economics developed is uni-dimensional, valid, highly reliable and fair. Thus, TAQE is of good quality and can now be used for assessment of student’s achievement in quantitative economics. Hence, it is a test instrument that can measure achievement in quantitative economics among secondary schools in Nigeria, and other countries. 7. 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The development of a diagnostic reading test of english for the students of medical faculty. Malang: Brawijaya University. Appendix A Test of Achievement in Quantitative Economics (TAQE) for Secondary Schools If the demand and supply equations for commodity X are represented by D = 10–1/3p; S = 9-1/4p. 1. Determine the equilibrium quantity for commodity X. (a) 12 (b) 4 (c) 10 (d) 6 Age distribution table of a State Age 0–16 17-35 36-65 66 and above Population 15000 10000 20000 5000 2. The ratio above indicates ____ dependency ratio. (a) low (b) high (c) equal (d) constant Use the following demand schedule to answer question 3. Price in Naira Quantity Demanded Old New Old New 25 35 100 80 3. Determine the percentage change in quantity demanded (a) 40% (b) 30% (c) 20% (d) 10% The population of 380 SS2 students in four schools is represented in percentages by the pie chart below. http://www.doscst.edu.ph/index.ph%5b/academics/graduateschool/publication/category/5-volum-1-issue-1-2012? https://scholar.google.com/scholar?hl=en&q=Student%20profiles%20and%20factors%20affecting%20performance http://dx.doi.org/10.1080/00207390150207095 http://psychology.about.com/od/researchmethods/ https://scholar.google.com/scholar?hl=en&q=Delicate%20balance%20between%20theory%20and%20practice:%20Health%20status%20assessment%20and%20item%20response%20theory http://dx.doi.org/10.1097/01.mlr.0000064780.30399.a4 http://www.africaeconomicanalysis/ http://www.eco.uc3m.es/temp/dolado/dolado_morales_ie_fv.pdf https://scholar.google.com/scholar?hl=en&q=Development%20and%20validation%20of%20diagnostic%20economics%20test%20for%20secondary%20schools http://dx.doi.org/10.5430/wje.v6n3p90 https://scholar.google.com/scholar?hl=en&q=Comparison%20of%201‐,%202‐,%20and%203‐Parameter%20IRT%20Models https://scholar.google.com/scholar?hl=en&q=Comparison%20of%201‐,%202‐,%20and%203‐Parameter%20IRT%20Models http://dx.doi.org/10.1111/j.1745-3992.1989.tb00313.x https://scholar.google.com/scholar?hl=en&q=Development%20of%20physical%20and%20mental%20health%20summary%20scores%20from%20the%20patient-reported%20outcomes%20measurement%20information%20system%20(PROMIS)%20global%20items https://scholar.google.com/scholar?hl=en&q=Development%20of%20physical%20and%20mental%20health%20summary%20scores%20from%20the%20patient-reported%20outcomes%20measurement%20information%20system%20(PROMIS)%20global%20items http://dx.doi.org/10.1007/s11136-009-9496-9 https://scholar.google.com/scholar?hl=en&q=Item%20response%20theory%20and%20health%20outcomes%20measurement%20in%20the%2021st%20century http://dx.doi.org/10.1097/00005650-200009002-00007 https://scholar.google.com/scholar?hl=en&q=Applying%20the%20rash%20model%20to%20the%20selection%20of%20items%20for%20mental%20ability%20test http://dx.doi.org/10.1177/0013164495055005006 https://scholar.google.com/scholar?hl=en&q=Construction%20and%20validation%20of%20achievement%20test%20in%20economics https://scholar.google.com/scholar?hl=en&q=Test%20item%20validity:%20item%20response%20theory%20(IRT)%20perspective%20for%20Nigeria https://scholar.google.com/scholar?hl=en&q=Construction%20of%20economics%20achievement%20test%20for%20assessment%20of%20students https://scholar.google.com/scholar?hl=en&q=Construction%20of%20economics%20achievement%20test%20for%20assessment%20of%20students http://dx.doi.org/10.5430/wje.v4n2p58 https://scholar.google.com/scholar?hl=en&q=Assessment%20of%20fit%20of%20item%20response%20theory%20models%20used%20in%20large-scale%20educational%20survey%20assessments http://dx.doi.org/10.1186/s40536-016-0025-3 Asian Journal of Education and Training, 2018, 4(1): 18-28 26 4 Generate the population of school D (a) 100 (b) 95 (c) 115 (d) 105 The population of five towns in Anambra State is shown below. Town Population 1 50 2 35 3 65 4 20 5 80 5 The percentage population of town 3 above is _______ (a) 20% (b) 26% (c) 36% (d) 30% Find the mean and median quantity of litres of fuel sold, using the frequency table below. Number of Litres sold Frequency 1 5 2 4 3 10 4 8 5 6 6 3 6 The mean quantity of the frequency table is ______ (a) 2.4 (b) 3.4 (c) 3.1 (d) 2.9 7 Calculate the lower quatile of the distribution above (a) 3 (b) 1 (c) 2 (d) 4 Individual demand schedule for coke bought by Mrs Ebo is shown below. 8 The curve of Mrs Ebo’s demand schedule has _____ slope. (a) lower (b) higher (c) nagative (d) positive Use the diagram below to answer questions 12 and 13 Price per bottle N Total quantity bought Per week 100 6 90 10 80 15 70 20 60 25 50 30 Asian Journal of Education and Training, 2018, 4(1): 18-28 27 9 At 0P2, FG illustrates _______ (a)excess demand (b)quantity supplied (c)equilibrium quantity (d) excess supply 10 D1D1 in the diagram above implies (a) increase in demand (b) decrease in demand (c) increase in quantity demanded (d) decrease in quantity demanded 11 If the price of a litre of fuel was N50 in 2016 but rose to N80 in 2017, the index number will be (a) 150 (b) 140 (c) 160 (d) 175 12 The answer above justifies that the value of money (a)fell by 62.5% (b) fell by 61.5% (c)rise by 62.5% (d) rise by 61.5% Use the table below to answer questions 14 and 15. Output of rice (kg) Total revenue N Marginal revenue N Total cost N Marginal cost N 10 150 - 250 - 20 200 5 300 5 30 350 15 430 13 40 450 F 500 7 50 550 10 550 M 60 600 5 580 3 70 630 X 700 12 13 Find the values of X and M (a)5 and 5 (b) 7 and 3 (c) 10 and 5 (d) 3 and 5 14 What will be the profit maximising output of this firm? (a) 10 (b) 30 (c) 50 (d) 70 A country produced 12 tons of goods at 2005 market price and in 2010 the country’s output remains 12tons, assuming the price rose by 50% in 2010. Using 2005 as the base year, 15 Calculate the price index for 2010 (a) 150 (b)100(c) 120(d) 200 16 Calculate the norminal value of the output in 2005 and 2010 (a) N1200 and N1500 (b) N1200 and N1200 (c) N1000 and N1200 (d) N1000 and N1500 17 Derive the real value of the output in 2010 (a)N1800 (b) N1200 (c) N1000 (d) N1500 If the weekly income of a consumer increased from N15 to N20, and quantity of goods bought per week increased from 10 to 15. 18 The income elasticity of demand for this consumer is ________ demand. (a) inelastic (b) elastic (c) unitary elastic (d) zero elastic If the price of commodity Y increased from 30k to 35k per unit and the quantity of another commodity X bought increased from 120 to 150. 19 The demand for commodity X is ___________demand. Asian Journal of Education and Training, 2018, 4(1): 18-28 28 (a) elastic (b) inelastic (c) unitary elastic (d) fairly inelastic Use the table below to answer question 29 and 30 Item N million Final consumption expenditure 500 Gross private investment 150 Government purchases 300 Depreciation 25 Net factor income -10 20 The net capital formation is (a) N115m (b) N125m (c) N915m (d) N925m Appendix B Test of Achievement in Quantitative Economic (TAQE) Answer (key) for Twenty (20) Multiple Choice Test Items ITEM KEY ITEM KEY NEW OLD NEW OLD 1 1 D 11 19 C 2 3 A 12 20 A 3 4 A 13 21 D 4 5 D 14 22 C 5 12 B 15 25 A 6 13 B 16 26 A 7 14 C 17 27 B 8 16 C 18 28 B 9 17 D 19 29 A 10 18 A 20 30 C Asian Online Journal Publishing Group is not responsible or answerable for any loss, damage or liability, etc. caused in relation to/arising out of the use of the content. Any queries should be directed to the corresponding author of the article.