Application of Chi-Square and T-Test in Architectural Research Methods

Transcription

1 Open Science Journal of Mathematics and Application 2016; 4(5): ISSN: (Print); ISSN: (Online) Application of Chi-Square and T-Test in Architectural Research Methods Isiwele A. Joseph *, Aikpehae A. Moses, Adamolekun M. Olusegun Department of Architecture, Faculty of Environmental Studies, Ambrose Alli University, Ekpoma, Nigeria address (Isiwele A. J.) * Corresponding author To cite this article Isiwele A. Joseph, Aikpehae A. Moses, Adamolekun M. Olusegun. Application of Chi-Square and T-Test in Architectural Research Methods. Open Science Journal of Mathematics and Application. Vol. 4, No. 5, 2016, pp Received: December 11, 2017; Accepted: January 3, 2017; Published: June 12, 2017 Abstract After data sorting, organizing and summarization, selecting and performing the most appropriate statistical analytic technique for testing and addressing the problem become next. Usually the method of analysis chosen will depend on the complicity of the research question as well as the level of measurement the data scores can be said to have attained with the constant variables. In this paper, two different methods of data analysis; the Chi-square and Student t-test, and their application in the field of architecture were discussed. Each of the tests was examined and its suitable uses stated with examples relevant to the field of architecture using secondary data. It was shown that each of the statistical methods has its unique application and can be used in the field of architecture. It is therefore recommended that architecture researchers and instructors take note and be equipped with the knowledge and application of these statistical analytical methods. This study therefore recommends the need for statistical methods in the field of architecture. Keywords Chi-Square, Student T Test, Statistical Methods, Architecture, Research 1. Introduction Architecture and indeed, the house, is often regarded and thus described as an encapsulation of the cultural heritage of a people [1, 2]. Over the past decade, there has been a particularly lively debate in architecture and allied fields about the extent to which design is or should be a template or more broadly perhaps, a new paradigm for research in creative or professional domains [3]. Architectural academicians have taken a notable diverse set of position and in discussing the essential role of research in architecture, Kieran [4] explicitly describes the relationship between design and research as essentially divergent but complementary. Research is a systematic inquiry directed toward the creation of knowledge [5]. Remarkably, this brief definition remains entirely consistent with characterization of research in contemporary architectural discourse. In the architectural context, analysis of research makes the case that research can be understood as any systematic inquiry, or as the close study of something [6]. Just as design can alternatively be understood as both a rational problemsolving technique or (sic) intuitive aesthetic act, research can be embodied in multiple mode of inquiry [3]. After sorting out, organizing and summarizing data, there is the need to select the most statistical analytic technique for testing of the hypothsis for purpose of addressing the problems of study as had been envisaged. According to Nayak and Hazra [7], most of us are familiar to some degree with descriptive statistical measures such as those of central tendency and those of dispersion but are falter at inferential statistics. In order to be able to make inference about population, the data type is very important. If a data scores achieve a measurements at the interval or ratio-scale level, the analytic method employed is parametric test. On the other hand, if the data sets achieve only nominal or ordinal levels of measurement, then non parametric method of analysis is employed. Statistical analysis is defined according to National Open University of Nigeria [8] as the refinement and manipulation

2 29 Isiwele A. Joseph et al.: Application of Chi-Square and T-Test in Architectural Research Methods of data in order to prepare them for the application of logical inference. Agbadudu [9] defines statistic as a numerical number that describe the sample. This includes the sample mean, variance, proportion. Note that both populations and samples can be described by stating their characteristics (numerical characteristics) and this is known as parameter(s) for a population. The characteristics of a sample given in the form of some summary measure are called statistics. It was then stated that for purposes of analysis, it is important that researcher makes clear distinction whether the objective will be fully met merely by: (i) classification of data into like groups; or to proceed further, (ii) uncover, establish or replicate relationship (whether correlative or causal or otherwise or (iii) attempt to develop new, adapt known techniques. This prior decision is important as it will help the researcher to employ appropriate techniques to handle data analysis most effectively. It is the aim of this paper to examine some basic statistical analysis, discuss bases for their uses and perform simplified examples in relation to the field of architecture. The statistical analysis of interest includes: Chi-square and the Student t-test. 2. The Chi-Square Test The chi-square; symbolically written as X 2 and pronounced as ki-square is a statistical tool that shows and or determines if there is relationship between two variables in the total population [10]. It is non-parametric statistics use in form of frequencies and is sometimes referred to as count data [11]. In another words, it is use in problems where information is obtained by counting rather than measurements. For example, how many houses passed the housing standard test, how many people lived in a particular type of house, how many building are supervised by architecture, how many story building are in a community e.t.c. As count data, chi-square method provides a powerful test for determining the significance of count data. Very often we are interest in knowing whether the observed frequencies of an event differ significantly from the expected frequencies. The chi-square test is useful in such situation. Under this situation the test becomes known as goodness- of-fit test Conditions for the Application of Chi-Square Test According to Kothari [11], the following conditions must be satisfied before test can be applied (i) Observations recorded and used are collected on a random basis, (ii) All the items in the sample must be independent, (iii) No group should contain very few items (Say less than 10, however, in cases where the frequencies are less than 10, regrouping is done by combing the frequencies of adjoin groups so that the new frequencies become greater than 10. Some statistician take this number as 5, but 10 is regarded as better by most of the statisticians), (iv) The overall number of items must also be reasonable large (It should normally be at least 50 howsoever small the number of groups may be), (v) The constraints must be linear constraints which involve linear equations in the cell frequencies of a contingency table (i.e. equation containing no squares or higher power of the frequencies are known as linear equation) Steps Involved in Applying Chi-Square Test The various steps involved in the application of chi-square test in this paper are as described by Kothari [11] and they are stated here as follows: Step i: First of all calculate the expected frequencies on the basis of given hypothesis or on the basis of null hypothesis. Usually in case of a 2 by 2 or any contingency table, the expected frequency for any given cell is worked out as under. Expected frequency of any cell = (Row total for the row of that cell x column total for the column of that cell)/ (Grand total) Equation 2.1 Step ii: Obtain the difference between observed and expected frequencies and fine out the squares of such difference i.e. calculate Observed (O) Expected (E) = (O - E) 2 Equation 2.2 Step iii: Divide the quantity (O - E) 2 obtained above by the corresponding expected frequency. i.e. (O E) 2 / (E) Equation 2.3 Step iv: Find the summation of (O - E) 2 / (E) values i.e (O - E) 2 / (E). Equation 2.4 This is the require value. The value obtained as such should be compared with relevant table value of and then inference is drawn The Level of Significance or Rejection The level of significance is often set at 0.05 (5%). For instance when the significance or rejecting level is set at 5% (0.05), it means there are five chance in a hundred that the data being investigated could have occurred by chance. The significance or rejection level is a measure of how strong the evidence must be before the null hypothesis is rejected or accepted. The significance level is usually 0.01 levels or 1%. This is because the adoption of 0.01 level of significance can generate a higher confidence in rejecting or accepting the null hypothesis Degree of Freedom for Chi-Square Test Degree of freedom is represented as df. The formulae for the calculation of df is df = (r-1) (c-i) Equation 2.5 Where r = number of rows and c = number of columns.

3 Open Science Journal of Mathematics and Application 2016; 4(5): Work Example One hundred and forty-four houses were sampled in an urban community with the following number of story buildings Number of story building bungalow a story building two story building Frequency/ number of count Is there a significant different in the number of story buildings in the urban community? Let us take the hypothesis (Ho) that there is no significant different in the number of story building in the urban Table 1. Working Chi-square table. community. The probability of obtaining one of the expected frequency of any number of story building is 144 x 1/3 = 48. Now we can write the observed frequencies and work out the value of as follows: Story building Observed Frequency (O) Expected Frequency (E) (O - E) (0 - E) 2 Equation 2.2 (0 - E) 2 /E Equation 2.3 Bungalow story story Applying equation 2.4 i.e (O - E) 2 / (E) = = Applying equation 2.5 to find the degree of freedom in the given problem is (r-1)(c-1) = (2-1)(3-1) = 2 Therefore, we find the table value of for 2 degree of freedom at 5 percent level of significance. This is 5.99 (see chi-square table [12]). The rule is that when the calculate value is greater that the tabulated value, we fail to accept the null hypothesis. Now comparing calculated value of (43.16) and table value of (5.99), it is observed that table value is less than calculated value. Thus, p is less than 0.05 indicating a significant different. This result thus fails to accept the null hypothesis. Therefore, there is a significant different between the number of story buildings in the sampled urban community. 3. The Student T-Test The student t-test statistic was said to be named student which was the pen name of William Sealy Gosset, a chemist who applied the test when he was working for the Guinness brewery in Dublin, Ireland in Richard [13], O'Connor and Robertson [14] and Fisher [15] provide the story and history of the introduction of the Student t statistic.a t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis. It can be used to determine if two sets of data are significantly different from each other. The t- test is a statistical test for testing the mean of population [16]; when the population is normally (or approximately normally) distributed. Among the most frequently used t-tests are: (1). Onesample test; whether the mean of a variable has a value specified in a null hypothesis and (2). Two-sample test; where means of two variables are compared. The two-sample t-tests for a difference in mean involve independent samples (unpaired samples) and the dependent sample (paired sample). The independent samples t-test is used when two separate sets of identically samples are obtained and compared. For example, let s assume we want to compare the strength of a new cement product (X) from the one we are using before (Y). In this case, we have two independent samples and would use the unpaired form of the t-test. On the other hand, the paired samples t-tests compare sample of matched pairs or a variable with repeated measures. For typical example of the paired samples t test would be observed if we assume we measure a room before a building commences (X 1 ) and at the completion of the building (X 2 ). According to John [17], the paired t-tests have greater power than unpaired tests as paired t-test reduces the effects of confounding factors Conditions for the Application of the Student T-Test The assumptions underlying a t-test are that (1) X (sample data) follows a normal distribution with mean (µ) and variance (σ 2 population standard deviation). (2) s 2 (the ratio of sample standard deviation) follows a χ 2 distribution (chi square distribution) with p degrees of freedom under the null hypothesis, where p is a positive constant and (3) Z and s are independent. These conditions are consequences of the variables being studied and the data sampled. For example, in the t-test comparing the means of two independent samples, the following assumptions should be met: (1) Each of the two variables being compared should follow a normal distribution. (2). The two variable being compared should have the same variance. Most two-sample t-tests are robust to all but large deviations from the assumptions [18] Steps Involved in Applying Student T Test The formula for the t-test is similar to that of the Z- test except that since the population standard deviation is unknown the sample standard deviation S is used Thus t =!/ # Equation 3.1 Where 0 = sample mean, u = hypothesized population mean, s = sample standard deviation and n = sample size

4 31 Isiwele A. Joseph et al.: Application of Chi-Square and T-Test in Architectural Research Methods However, in comparison of two independent groups, t becomes [12] t = (X 1 X 2 ) / SE(X 1 X 2 ) Equation 3.2 Where X 1 - X 2 = the difference between the mean of the first and second samples, SE(X 1 - X 2 ) =Standard error of the difference between the two mean values. The calculation of the standard error may be complicated for the less mathematically minded person. Thus, SE(X 1 X 2 ) can be determine using this formula [12] SE(X1) = $ %& #& = S 1/ '1 Equation 3.3 This is simply the standard deviation of the measurement divided by the square root of the sample size Thus, SE(X 1 X 2 ) = $ %& #& + % # or (S 1/ '1+ S 2 / '2) Equation 3.4 Therefore, substituting the equation of SE(X 1 ) in equation 3.2, t will therefore be Or T = (X1 X2) / $ %& + % #& # Equation 3.5 T = (X1 X2) / (S 1 / '1+ S 2 / '2) Equation Degree of Freedom for Student T-Test The degree of freedom for the t test is given as (n n 2-1). This is similar to (n 1 + n 2 2). Here n1 and n2 represent the sample size groups 1 and Work Example 1 For example, a machine is designed to fill bags with 50Kg of cement. A builder suspects that the machine is not filling some of the bags completely. Samples of 30 bags were then collected and give a mean of 46Kg and a standard deviation of 5.3Kg. The problem is to find if there is enough evidence to support the customer s inference. The first thing to do is to formulate hypothesis Here null Hypothesis (Ho) states that u 50kg and alternated hypothesis (Ha) states that u 46kg (which is the builder s claim). In another words, the null Hypothesis states that there is no significant different in the bags the machine filled while the alternate hypothesis states that there is a significant difference in the bags filled by the machine. Applying equation 3.1 since it a one sample t we have t = 0 u S/ ' /0123 = 2 = 2 4 / 0.968= /.5/ 50 /.5/ The degree of freedom for the equation above is = (n 1) = (30 1) = 29. A check at the t-table at 5% level of significance (ie 95% confidence interval), the table value at 29 df = 2.05 (critical value) but the calculated value is The rule is that when the calculate value is greater that the tabulated value, we fail to accept the null hypothesis and say that there is a significant different. Thus we reject the null hypothesis and thus p value is less than Since the test value falls within the critical rejection region Ho: is rejected and instead it is held that there is enough evidence to support the claim that the machine is not filling the bags with 50 grams of the required cement Work Example 2 The mean time in days taken for 10 men to build a story building to the lentel level in Ekpoma is 21 days with a standard deviation of 5 days. Another comparable 10 men in Auchi will build similar story building to the lentel level in 28 days with a standard deviation of 6 days. Is there a significant difference in the number of days taken as a result of the location different? Note that this is a two independent sample groups so we apply equation 3.2. But first we need to state the Ho and Ha. Ho = there is no different in the number of days taken to build between the two groups or X 1 X 2 = 0. On the other hand, Ha = there is a different in the number of days taken to build between the two groups or X 1 X 2 0. Thus, using the formula t = (X 1 X 2 ) / SE(X 1 X 2 ). Equation 3.2 but standard deviation is what was given. Therefore, the formula changes to t = (X1 X2) / (S 1 / '1+ S 2 / '2) Equation 3.6 t = (28 21) / (6/ / 10) t = (7) / (6 + 5 / 10 ) t = 7 x 10 / 11 t = / 11 t = this is t calculated. The degree of freedom for question above is = (n1 1+ n2 1) = ( ) = 18. A check at the t-table at 5% level of significance (ie 95% confidence interval), the table value at 18 df = 2.10 (critical value) but the calculated value is The area to the right of the critical value is called critical region or rejection region and since the calculated t value is to the rejection region, we reject the Ho hypothesis. Thus, there is a significant different in the number of days taken to build between the two locations. 4. Conclusion Groat and Wang [3] has previously published a text on research methods specifically aimed at architectural scholars

5 Open Science Journal of Mathematics and Application 2016; 4(5): and practitioners. According to Groat and Wang [3], with the growth in pressure on academic staff and the increasing importance of research postgraduate qualifications in the architectural profession and in related specialist areas in architectural schools around the world improving research performance in the profession is timely. It article showed that each of the statistical methods herein presented has its unique application and can be used in the field of architecture. Thus, we recommend that architecture researchers and instructors take note and be equipped with the knowledge and application of these statistical analytical methods and others. While this is not only useful for architectural fellows on training and learning research skills, it is important that fellows who have already developed research skills acquire statistical knowledge too. References [1] Rapaport, A. (1969); Houseform and Culture, Prentice Hall, New Jersy. [2] Kalilu ROR (1997). The House as an Encapsulation and Metaphor of Life: New Theoretical Perspectives in Nigerian Architecture in Amole B. (ed): The House in Nigeria, Proceedings of National Symposium Held in Obafemi Awolowo University, Ile-Ife, Nigeria, 23rd 24th July. Pp [3] Groat, L. and Wang, D. (2002). Architectural Research Methods. Reviewed by Kerry London and Michael Ostwald. New York: John Wiley and Sons. [4] Kieran, S. (2007). Research in design: Planning doing monitoring learning. J. Architectural Education; 61(1): 31. [5] Snyder, J. (1984). Architectural research. New York: Van Nastrand Reinhold. Pp 2. [6] Salomon, D. (2011). Experimental cultures: On the end of the design thesis and the rise of the research studio. J. Architectural Education; 65(1): [7] Nayak K. B. and Hazra, A. (2011). How to choose the right statistical test? Indian J Ophthalmol.; 59(2): [8] National Open University of Nigeria (2012). Research methods in criminal justice. Course code: POS 411. School of arts and social sciences. Pp [9] Agbadudu, A. B. (1994). Statistics for business and the social sciences. Benin City: Uri Publishing Ltd. [10] Agbonifoh B. A. and Yomere G. O. (1999) Research Methodology in the Management and Social Sciences Uniben Press University of Benin, Benin City, Nigeria. [11] Kothari, C. R. (2004). Research methodology; methods and techniques. New age international (p) limited publishers. 2 nd revised edition. [12] Bamgboye, E. A. (2007). A companion of medical statistic. 3 rd Edition. IBIPRESS and Publishing Co. Ibadan, Nigeria. Pp 205. [13] Richard M. (2004). The Story of Mathematics (Paperback ed.). Princeton, NJ: Princeton University Press. p [14] O'Connor, J. J. and Robertson, E. F. (1987). "William Sealy Gosset", MacTutor History of Mathematics archive, University of St Andrews. [15] Fisher Box, Joan (1987). "Guinness, Gosset, Fisher, and Small Samples". Statistical Science. 2 (1): [16] Bluman, G. A. (1998). Elementary Statistics: A step by step approach. 3 rd Edition. The McGraw Hill companies. [17] John A. Rice (2006), Mathematical Statistics and Data Analysis, Third Edition, Duxbury Advanced. [18] Martin (1995). An Introduction to Medical Statistics. Oxford University Press. p. 168.