Quantitative Research Methods in Political Science Lecture 12 (12/05/2024) PDF

Quantitative Research Methods in Political Science Lecture 12: Course Review Course Instructor: Michael E. Campbell Course Number: PSCI 2702 (A) Date: 12/05/2024  Introduction This lectur...

Quantitative Research Methods in Political Science Lecture 12: Course Review Course Instructor: Michael E. Campbell Course Number: PSCI 2702 (A) Date: 12/05/2024  Introduction This lecture includes information about many of the foundational concepts covered in this course, based on questions submitted by students It does not include all the material that will be on the final exam Please see the study sheet that was distributed, additional PowerPoints, and assigned readings for full information Furthermore, it will also benefit you to consider the material that was on your assignments  Introduction Cont’d Research is “any process by which information is systematically and carefully gathered for the purpose of answering questions, examining ideas, or testing theories” (Healey, Donoghue, and Prus 2023, 10) What separates quantitative research from qualitative research is that they make use of statistics… We can understand statistics as a set of mathematical techniques used to organize and manipulate data (think of them as tools) We use these tools to “analyze data, to identify and probe trends and relationships, to generalize, and to revise and improve our theories” (Healey, Donoghue, and Prus 2023, 16).  Natural vs. Social Science Quantitative methods are based on the use of the scientific method: an organized series of steps that help researchers study empirical reality Scientific method was originally developed to study the natural world There is more ‘noise’ in the social world – i.e., there are several different things that occur in the social world that might affect variation in our results Consequently, we cannot control for things in the same way when we are conducting research in the social world We can take some of these things into account (i.e., control variables), but we can never consider them all  The Role of Statistics in Social Science Research When we conduct quantitative research, it can be visualized as a broad process in which: (a) researchers have a theory… (b) they develop hypotheses (c) they make observations (d) and then they make empirical generalizations  Variables Variables are any trait that changes in value from case-to-case At their most basic definitional level, this means they are things that vary Variables have three characteristics (1) Mutual (2) (3) Homogeneity Exclusivity Exhaustiveness  Mutual Exclusivity Response categories must be mutually exclusive Not Mutually Mutually Exclusive Exclusive This means that each 18-24 18-24 response category in a 23-34 25-34 variable must be clearly defined and not overlap 33-44 35-44 45-54 45-54 Above 54 Above 54 For example, “What is your age?”  Exhaustiveness Response categories must be exhaustive Note Exhaustive Exhaustiv This means the variable e must encompass all Red Red possible categories or Blue Blue values, ensuring every Green Green potential outcome is covered Yellow Yellow Other For example, “what is your favorite color?”  Homogeneity Response categories should be homogenous Not Mutually Mutually Exclusive This means that the Exclusive categories within the Ford Ford variable should be Suzuki Suzuki consistent in nature, each Honda Honda measuring the same characteristic or attribute Tomato Other Other For example, “what type of vehicle do you drive?”  Discrete and Continuous Variables There are two overarching types of variables: 1. Discrete 2. Continuous Discrete variables: these are variables whose basic units cannot be subdivided Continuous variables: these are variables Depending on the type of thing you are that can be subdivided infinitely studying, if it is a discrete variable, it will always be a whole number For example, time is a continuous variable, For example, if you are measuring the because it can be measured in weeks, members of a household, it can never not months, days, hours, minutes, seconds, be a whole number… milliseconds, and so on… A household wouldn’t have 1.7 or 2.5 For example, let’s say we observe the time it people living in it, it would have a whole number takes people to complete a race… One person might run it in 1 hour, and another might run it in 0.50 hours  These are the mathematical nature of variables under consideration Levels of There are three levels of Measurement measurement: 1. Nominal 2. Ordinal 3. Interval Nominal Level Variables Area Frequenc This is the lowest level of Code y measurement 613 10 753 22 The response categories associated 683 15 with these variables cannot be ranked 437 19 Responses are nothing more than 365 7 labels (even if they are numerical in nature) Other 12 Ordinal Level Variables These are ranked variables They are more precise than nominal Response Frequency variables Very good 10 Somewhat good 25 In addition to classifying them into categories, we can describe the Neither good nor 7 categories as “more or less” or “higher bad or lower” Somewhat bad 27 Very bad 36 But we cannot distinguish the exact distances between scores For example, Satisfaction with PSCI 2702 could be an ordinal variable if survey respondents were asked to rank their experience on a scale from “Very  Interval-Ratio Level Variables This is the highest level of measurement (and the most precise) Can be used to identify exact distances between scores (because there is an equal distance between scores) For example, family size: the difference between 1 and 2 children is 1, the difference between 2 and 4 children is 2, and so on… Difference between “interval” and “ratio” is that ratio level variables have a naturally occurring zero – e.g., an absence of income, an absence of children, etc… Temperature is an “interval” level variable, because a temperature of zero does not mean that temperature does not exist Characterist ics of Levels of Measureme nt  Conceptualization and Operationalization Before we can work with variables, you need to understand what these variables represent… When we start research, we usually begin with concepts A concept is: an idea or mental construct that organizes, maps, and helps us understand phenomena in the real world and make choices. For example, what is a political party? It can mean many things (see next slide)…  Example: ‘Political Party’ as a Concept Political Thinker Definition Edmund Burke (1770) “[A] party is a body of men united, for promoting by their joint endeavors the national interest, upon some particular principle in which they are all agreed.” Anthony Downs (1957) “[A] political party is a coalition of men seeking to control the governing apparatus by legal means. By coalition, we mean a group of individuals who have certain ends in common and cooperate with each other to achieve them.” V.O. Key, Jr. (1964) “A political party, at least on the American scene, tends to be a “group” of a peculiar sort…Within the body of voters as a whole, groups are formed of persons who regard themselves as party members…In another sense, a “party” may refer to the group of more or less professional workers…At times party denotes groups within the government….Often it refers to an entity which rolls into one the party-in the-electorate, the professional group, the party-in-the-legislature, and the party-in-the-government…In truth, this all encompassing usage has its legitimate application, for all the types of groups called party interact more or less closely and at times may be as one. Yet both analytically and operationally the term ‘party’ most of the time must refer to several types of group; and it is useful to keep relatively clear the meaning in which the term is used.” William Nisbet “[A] political party in the modern sense may be thought of as a relatively durable social formation which seeks offices or power Chambers (1967) in government, exhibits a structure or organization which links leaders at the centers of government to a significant popular following in the political arena and its local enclaves, and generates in-group perspectives or at least symbols of identification or loyalty.” Leon D. Epstein (1980) “[What] is meant by political party [is] any group, however loosely organized, seeking to elect government office holders under a given label.” Ronald Reagan (1984) “A political party isn’t a fraternity. It isn’t something like the tie you wear. You band together in a political party because of certain beliefs of what government should be” Robert Huckshorn “[A] political party is an autonomous group of citizens having the purpose of making nominations and contesting elections in (1984) hope of gaining control over governmental power through the capture of public offices and the organization of the government.” Joseph Schlesinger “A political party is a group organized to gain control of government in the name of the group by winning election to public (1991) office.” Even if a concept seems simple, it can nevertheless mean many different John Aldrich (1995) “Political parties can be seen as coalitions of elites to capture and use political office. [But] a political party is more than a  Conceptualization and Operationalization Cont’d Concepts can be more or less complex… For example, what do political participation or social status refer to? Hard to say, because they are ideas Concepts are essentially large, broad ideas, that reflect ambiguously defined phenomena  Conceptualization and Operationalization Cont’d Question: “does globalization exacerbate climate change?” (see PowerPoint #2) How do we know globalization and climate change when we see them? Does globalization refer to increased trade? Or does it refer to greater levels of transnational political convergence? Does climate change refer to rising sea levels? Or does it refer to variations in temperature? Therefore, the question stated above is a conceptual question: “a question expressed using ideas, is frequently unclear and is difficult to answer empirically” (Pollock and Edwards 2020, 2).  Conceptualization and Operationalization Cont’d To answer a conceptual question, one needs to conceptualize and operationalize… This will lead to the development of a concrete question: “a question expressed using tangible properties, [which] can be answered empirically” (Pollock and Edwards 2020, 2). Our goal, in empirical research, is to transform conceptual questions into concrete ones To do this, we must transform concepts into concrete terms so that ambiguous terms can be described and analyzed (through the process of conceptualization and operationalization)  When dealing with a concept, we must first develop a conceptual definition… A Conceptual Definition: “clearly describes the concept’s measurable properties and specifies the units of analysis (e.g., people, nations, states, and so on) to which the concept applies.” Conceptual In other words, researchers develop conceptual definitions because they are more precise and clear Definition expressions of what they are talking about… But this definition can only be written once we have settled on a set of properties that best represent the concept A conceptual definition communicates the subjects to which the concept applies and suggests a measurement strategy  A conceptual definition leads to an operational definition: “describes the instrument to be used in measuring the concept and putting a conceptual definition “into operation”” (Pollock and Edwards 2020, 8). An operational definition is a precise indication of how the concept is measured – i.e., it indicates Operational how we know there is more or less of a phenomenon Definition For example, let’s say your concept was height and you wanted to measure height – your operational definition might be “height” as defined by the number of feet/inches a person is tall. These are usually accompanied by an instrument  Process of Conceptualization and Operationalization The process of conceptualization follows a series of systematic steps: 1. Clarify the concept: make a list of the concept’s concrete properties – i.e., the characteristics that best represent the concept These properties must be concrete (i.e., they must be observable) These properties must vary (i.e., they must occur or not occur, or occur at varying levels) Once you have identified these characteristics, you must select the characteristic that best defines the concept (but also the one that makes the most sense in the context of your research)  Process of Conceptualization and Operationalization Cont’d 2. Develop a Conceptual Definition A conceptual definition must communicate three things: (1) the variation within a measurable characteristic or set of characteristics (2) the subject or groups to which the concept applies (3) how the characteristic is to be measured For example, if we selected “trade and transactions” between countries as the most essential characteristic of globalization our conceptual definition would be: “The concept of (a) globalization is defined as the extent to which (b) countries exhibit the characteristic of (c) high levels of trade and transactions.”  Process of Conceptualization and Operationalization Cont’d By stating that (a) globalization is defined by the extent to which, it restated the broad idea being conceptualized, and it also pointed to the existence of variation, meaning that it can exist at varying levels, or not at all By identifying (b) countries, we identified to whom or what the concept applies (i.e., the unit of analysis) By stating that (c) high levels of trade, we identified the way the concept can be measured  Process of Conceptualization and Operationalization Cont’d 3. Operationalization When you operationalize, you are explicitly stating how a concept will be measured empirically For instance, how would we know if trade and transactions occur at a high or low level? How would we measure the level of trade and transactions among countries? We could develop an operational definition that reads: “The total amount of imports and exports of a country, in millions of current year US dollars.”  Process of Conceptualization and Operationalization Cont’d 4. Collect Data or Select Relevant Variables Please respond to the following. On a scale of 1 to 10, with 10 being Once you have clarified your concept and Very High, and 1 being Very Low, developed conceptual and operational rate your shopping experience at definitions, you can begin to collect data Loblaws… 1 – Very Low You would do this by creating a survey and 2 asking people questions using your 3 instruments to measure the concept… 4 5 For instance, let’s say you wanted to know peoples’ shopping experience at Loblaws, 6 you might provide this… 7 8 9  Process of Conceptualization and Operationalization Cont’d 4. Collect Data or Select Relevant Variables In most cases, though, we do not have the resources to collect our own data… So, we might compile data from across various sources and literature Or you can find a dataset that contains information on your operationalized concept, and you can select the variables that are most appropriate for your research (as you did in Assignment #1)  Descriptive vs. Inferential Statistics Once you know what type of variables you’re working with, and you have data available to you, you can examine many things (but what you examine depends on your use of descriptive or inferential statistics) Descriptive statistics are essentially used to: 1. summarize or describe the distribution of a single variable – these are called univariate descriptive statistics 2. describe the relationship between two or more variables – these are called bivariate or multivariate descriptive statistics (e.g., measures of association) Inferential statistics, on the other hand, are different from descriptive statistics, in that they allow us to generalize from our sample to our population… In other words, inferential statistics allow us to use information about our sample to make inferences about the larger population from which the data was drawn  Descriptive vs. Inferential Statistics Cont’d Univariate descriptive statistics are essentially a starting point for statistical analysis… If you have a variable with 126 observations, it will take too long to look at each one… But we don’t need to do that, because we have descriptive statistics These statistics include proportions, percentages, rates, and ratios… Whereas proportions and percentages are used to standardize raw data and compare parts of a whole and can also be used to compare groups of different sizes Rates and ratios are used when we want to compare categories of a variable in terms of relative frequency or to summarize the distribution of a single variable  Measures of Central Tendency and Dispersion are also types of descriptive statistics… They “reduce huge arrays of data to a single, easily understood number” (Healey, Donoghue, Measures Prus 2023, 80). of Central Measures of Central Tendency: allow us to describe data in such a way that we can Tendency identify the typical or average case in a distribution Therefore, when we are talking about the measures of central tendency, we are talking about statistics that can help us summarize data in such a way that we can describe its most common scores, the middle case, and the average of all cases combined  Measures of Central Tendency 1. Mode: this is the most recurring score in a distribution of data and is most often used when working with nominal level data (in fact it is the only measure of central tendency that can be used with nominal level data) 2. Median: this is the exact center of a distribution, meaning exactly half of all cases will fall above the median, and half of all cases will fall below the median (this is most often used with ordinal level data) 3. Mean: this reports the average score in a distribution and is most commonly computed for interval-ratio level data It is the score that comes as close to every score in a distribution It is always at the center of any distribution Unlike the median, it is the point around which all of the scores cancel out  Measures of Dispersion Measures of central tendency need to be combined with Measures of Dispersion to completely describe data… Measures of dispersion provide information about “the amount of variety, diversity, or heterogeneity within a distribution of scores” (Healey, Donoghue, and Prus 2023, 80). In other words, they tell us about the way the data are spread out… Data dispersion refers to the spread of data (i.e., data can be more or less dispersed) (See right)  Measures of Dispersion Cont’d The best measures of dispersion will: When we use data measured at different levels, we use different measures of dispersion: 1. use all of the scores in a distribution, meaning the statistic will be computed using all of the information that is 1. The Mode = Index of Qualitative available Variation 2. describe the average or typical deviation of the scores and give us an 2. Median = Range or Interquartile idea of how far the scores are from one Range (depending on presence of another and from the center of the outliers) distribution 3. Mean = Variance or Standard 3. increase in value as the distribution of Deviation scores becomes more diverse Measures of Dispersion Cont’d  The Normal Curve The Normal Curve is a theoretical model we use in statistics It is paired with the mean and standard deviation, allowing us to precisely describe empirical distributions It also bridges the gap between descriptive and inferential statistics… At its core, the normal curve is “a special kind of perfectly smooth frequency polygon that is unimodal (i.e., it has a single mode or peak) and symmetrical (unskewed) so that its mean, median, and mode are all exactly the same value” (Healey, Donoghue, and Prus 2023, 126)  The Normal Curve Cont’d It is theoretical, because you will never find data so perfectly distributed in reality… When measured using standard deviations, the distances along the horizontal axis will always encompass the same proportion of the total area under the curve This means when you use the normal curve, think in terms of standard deviations We use standard deviations, because the distance between any point and the mean cuts off the same proportion of the total area when measured in standard deviations  The Normal Curve Cont’d Example: you have a sample of 1000 people, and you are researching the amount of time they spend studying for a final exam in minutes… The data range from 40 (low) to 160 (high) The mean is 100 The standard deviation is 20  The Normal Curve Cont’d As you can see, for every standard deviation, you are moving up or down 20 minutes This means that the average person studies 100 minutes One standard deviation above the mean is 120 minutes One standard deviation below the mean is 80 minutes Two standard deviations above the mean is 140 minutes Two standard deviations below the mean is 60 minutes And so on…  Area under the Normal Curve When measured in standard deviations, distances along the horizontal axis on any normal curve will always encompass the same proportion of the total area under the curve Standard Deviations and the Area Under the Normal Curve Between Lies -1 and +1 standard 68.26% of the area deviation -2 and +2 standard 95.44% of the area deviations -3 and +3 standard 99.72% of the area  Area under the Normal Curve Cont’d The area under the normal curve can also be expressed in numbers of cases as opposed to the percentage of total area… For example, if you have a sample with 1000 cases, 683 will fall within 1 standard deviation from the mean This is because 68.26%, when rounded, represents 683 out of 1000 Therefore, 683 students spend between 80- and 120-minutes studying for final exams Remember, scores cluster around the Likewise, 954 students spend between 60- mean under a Normal Curve and 140-minutes studying (i.e., 95.44% of cases)  Area under the Normal Curve Cont’d Generally, we work with whole number values for areas under the curve… These are 90%, 95%, and 99% Each corresponds with specific standard deviation values Between Lies +/- 1.65 Std. Dev. 90% of area +/- 1.96 Std. Dev. 95% of area +/- 2.58 Std. Dev. 99% of area  When we use Z scores, we are converting our raw scores into standardized scores (measured in terms of standard deviations) With this information, you can find out exactly where a score falls under a normal curve (its distance from the mean, and the area beyond the Z Scores Z score) (Standard For example, if you have a Z score of +2, the Scores) proportion of area between it and the mean is 0.4772 Or, 47.72% of the area under the curve falls between the mean score and a Z sore of +2 Why? Divide 95.44% by 2 Remember, 2 standard deviations from the mean encompasses 95.44% of the area under the curve  Likewise, if you had a Z score of +1.96, it would Z Scores indicate that a proportion of 0.4750, or 47.5% of the area under the curve falls between the Z (Standard score and the mean… Scores) 5% of the area would fall beyond your Z score Cont’d This is because ±1.96 standard deviations will encompass 95% of all area under the curve  Samples and Sampling When we conduct research using quantitative methods, we often work with samples… This is because inferential statistics requires that we generalize from a sample to a population Samples must be representative (so we collect probability/random samples) The best way to collect a probability sample is to use EPSEM technique (Equal Probability of Selection Method) However, even if probability samples are representative, they will not be perfectly representative (known as sampling error)  Generally, we do not know much about our population of interest (this is why we use samples) We generally do not know its mean or standard deviation To link information from the sample to the population, we use a theoretical device known as the sampling distribution Sampling The sampling distribution is “a theoretical, probabilistic Distribution distribution of a statistic for all possible samples of a certain sample size (n)” (Healey, Donoghue, and Prus 2023, 156). This means that the sampling distribution includes statistics that represent every conceivable combination of cases (or every possible combination of samples of the same size) from a population. It is theoretical because the Sampling Distribution is based on the laws of probability, not on empirical information…  Sampling Distribution Cont’d Therefore, when we use inferential statistics, three distinct distributions are always involved… 1. The Sample Distribution: this exists in reality (i.e., it is empirical) and we know that the shape, central tendency, and dispersion of any variable can be known for the sample. Remember that the information from this sample is only important insofar as it allows the researcher to know about the population. 2. The Population Distribution: this also exists in reality (i.e., it is empirical), but information about it is unknown. Amassing information or making inferences about the population is the sole purpose of inferential statistics. 3. The Sampling Distribution: this does not exist in reality (i.e., it is non-empirical, or theoretical). However, using the laws of probability, we know a great deal about this distribution.  Sampling Distribution Cont’d The sampling distribution awards us the ability to estimate the probability of any sample outcome… This is because when you have a large enough sample, it means you can compute an infinite number of sample statistics (we looked at the mean this semester) Therefore, if you took repeated samples of the same size from a population, you would compute an infinite number of sample means (all of which would be slightly different from one another)  Sampling Distribution Cont’d But remember, EPSEM doesn’t guarantee that the sample we select will necessarily be representative of the population… At times, the sample means may be very high or very low (especially when compared to the true population mean) You know some outcomes are going to be very rare When we construct a sampling This Is because if you go back to the distribution, we sample into infinity concept of the normal curve, we know that Therefore, the number of “misses” (or the vast majority of observations are going means that are very far from the true to cluster around the mean population mean) are going to be equal on both sides of the mean  Theorems Underpinning the Sampling Distribution Two theorems underpin the Sampling Distribution: 1. If we begin with a trait that is normally distributed across a population and we take an infinite number of repeated random samples of an equal size from that population, then the sampling distribution of sample means will be normal in shape Since the sampling distribution is normally distributed, this also tells us that the mean of the sampling distribution will be the same as the population. This also tells us that the standard deviation of the sampling distribution is equal to the standard deviation of the population divided by the square root of the sample size (n) Therefore, we can estimate the mean and standard deviation of a population using sample statistics…  Theorems Underpinning the Sampling Distribution Cont’d However, the first theorem requires that the distribution of the population be normal in shape… But what if we don’t know? This is covered by the second theorem: The Central Limit Theorem The central limit theorem tells us that if a trait is not normally distributed in a population, we can still construct a normal curve if we increase the size of our sample This removes the constraint of normality in the population Theorems Underpinning the Sampling Distribution Cont’d Generally, the sample should be at least 100 in size for the Central Limit Theorem to apply Therefore, the sampling distribution will take on the shape of the normal curve as the sample size increases This means we can use our knowledge of the normal curve to make certain estimations and predictions…  Estimation Procedures Neither the sampling distribution nor the Normal Curve exist in reality… But we can use these theoretical concepts to collect a sample and make certain inferences based on that sample The first application for inferential statistics, using the concepts of the Normal Curve and Sampling Distributions are to compute: 1. Point Estimates: a sample statistic that is used to estimate the population value 2. Confidence Intervals: consist of a range of values (an interval) instead of a single point  Estimation Procedures Cont’d To compute either one of these, you need an estimator An estimator is something known about the samples – be it a mean, median, or mode For example, a sample mean () can be considered an estimator of the population mean (), the sample standard deviation (s) can be an estimator for the population standard deviation (), and so on… For an estimator to be of value it must be both: 1. Unbiased 2. Efficient  Unbiased Estimator An estimator is unbiased if, and only if, the mean of its sampling distribution is equal to the population value of interest We also know that under a normal curve 68.26% of all cases will fall within 1 If the sample size is large enough, it will take on standard deviation from the mean, 95.44% a normal shape – meaning it will have the same of all cases will fall within 2 standard mean as the population deviations from the mean, and 99.72% of all cases will fall within 3 standard deviations from the mean Remember, if you take an infinite number of samples of the same size, from the same population, the mean of each sample will be slightly different, but overtime, the mean of all Only very few cases fall beyond 3 standard these samples will equal the same as the deviations… population… Therefore, if an estimator is unbiased, it is Therefore: (1) the mean of a sampling distribution will be likely accurate in terms of estimating the the same as that of a population population parameter (in this case the (2) a sample size of a sufficient size (namely population mean ()) when samples are at least 100 in size) will take on a normal shape  Efficient Estimator Efficiency refers to the extent to which the sampling distribution clusters around the mean The smaller the standard deviation of a sampling distribution (a.k.a., the standard error), the more the sampling distribution clusters around the mean The standard error is an inverse function of the sample size – meaning that the standard error will decrease as sample size increases… Consequently, we can improve the efficiency of any estimator by increasing the sample size  Efficient Estimator Cont’d Let’s say you want to determine the average income of full-time workers in a community, so you take two different samples One sample has 100 respondents, the Sample 1 Sample 2 other has 1000 (see right) = $75 000 = $75 000 = 100 = 1000 Both sample means are unbiased, indicating the mean of the sampling distribution is the same as the mean of the population (as per the first theorem)  Efficient Estimator Cont’d Now, pretend you know the population standard deviation is $5000 If we calculate the standard error for each sample, based on their different sizes… For sample 1, where n = 100, the calculation would be or which gives us an SE of $500.00 For sample 2, where n = 1000, the calculation would be or which gives us an SE of $158.13 Therefore….  As you can see, the sampling distribution is much more clustered around the mean when n = 1000 Therefore, the estimate for this sample is therefore much more likely to be close in value to the population parameter than the other sample where n = 100 This is because, the larger the sample, the closer it is to reality  Point Estimate Based on all of that, you can construct a point estimate (it’s very simple) 1. Draw an EPSEM sample 2. Calculate the mean or proportion for your sample The mean of the sample is the point-estimate for the population (but it could also be the median or mode)  Point Estimate Cont’d In our example, we wanted to know how much money people make on average in a community And our sample mean was $75 000 So, we can say “full time workers in this community have an income of approximately $75 000” This is not to say that the actual mean of the population is $75 000 (remember that EPSEM samples aren’t always going to be representative – it is possible you selected a sample out in the tails… But we can be rather confident that our sample mean approaches the population mean because of the theorems that underpin the sampling distribution Furthermore, the larger the sample, the more confidence we can have in our estimator, because it will be more efficient and there will be less variability  Confidence Intervals are more complex than point estimates, but are safer… This is because you estimate a range of values into which the population parameter is likely to fall (as opposed to a single value) To calculate a confidence interval, you must first select Confidence an alpha level (most commonly 0.05) Intervals This is because 5% of the area under the curve will fall beyond an alpha of 0.05 When alpha is set at 0.05, you are constructing a 95% confidence interval (meaning you are willing to be wrong only 5% of the time) This is the likelihood of selecting a rare sample – i.e., selecting a sample that happens to fall in the tails under a normal curve  Confidence Interval Cont’d When you set alpha at 0.05 when constructing a confidence interval, your probability of error is divided equally into the upper and lower tails (0.025 in each) You then need to find Z scores associated with these alpha levels When you set alpha at 0.05, and each tail contains 0.025 it will always correspond with a Z score of 1.96  Confidence Intervals Cont’d Depending on what you set alpha at, different Z scores will be associated with the alpha (see right) However, if you do not know the standard deviation of the population, you must use the t distribution We are most interested in 0.05  Confidence Intervals Cont’d Now, we do not normally know the population standard deviation, so we need to use the t distribution here The t distribution accounts for the use of the sample standard deviation as opposed to the population standard deviation Its shape varies as a function of sample size – meaning the larger the sample, the more accurate our estimate will be As the sample size increases, the t distribution takes on a normal shape – i.e., as the degrees of freedom increase, the t and Z distributions converge Anything above 120 degrees of freedom means that the t and Z distributions will be equal This means that t scores and Z scores will be the same… Confidence Intervals Cont’d  Confidence Intervals Cont’d So, if you have a sample mean and sample standard deviation, but do not know the population standard deviation, you use this formula to compute the confidence interval: c.i. = -In this equation: = the sample mean t = the t score as determined by the alpha level and n – 1 degrees of freedom = the estimated standard error of the mean when is unknown  Confidence Intervals Cont’d Therefore, with a sample of 1000, and a standard error of 158.13, and a mean of $75 000… c.i. = c.i. = Notice t = 1.96 This is because the sample size is large enough that the t c.i. = distribution converged with the Z distribution c.i. =  Confidence Intervals Cont’d Therefore, based on your sample information, you can say that “if you were to take repeated samples of the same size (1000 in this case) from the population, 95% of those samples would contain the true population value…” You are 95% confident that the true population value (in this case the population mean) falls between $74 990.2 and $75 009.8 Therefore, people in the community from which you drew the sample make between $74 990.2 and $75 009.8  Two things affect the width of a confidence interval: 1. The level at which you set alpha 2. The sample size Controlling the Width The higher you set alpha, the wider your of Intervals confidence interval will be – because you are expanding the area under the curve into which the population parameter might fall Conversely, the greater the size of the sample, the smaller the width of the confidence interval – because your estimates will be more precise  Hypothesis Testing is the second application of inferential statistics A hypothesis is a statement about the relationship between two variables When you develop hypotheses, you will have a null Hypothesis hypothesis and a research hypothesis 1. Null Hypothesis: is a statement of “no difference” or “no Testing relationship” between variables (Symbolically or or ) relationship between variables (Symbolically > or or ) 2. Research Hypothesis: is a statement of difference or Our goal is to reject the null hypothesis… This is why hypothesis testing is defined as “an inferential statistical procedure designed to test for the relationship between variables, or a difference between groups of cases, at the level of the population” (Healey, Donoghue, and Prus 2023, 213).  Hypothesis Testing Cont’d We can also call hypothesis testing “significance testing” – because we are trying to find statistically significant results When we find statistically significant results, we can reject the null hypothesis at a pre-determined significance level (usually set at 0.05)… When we test hypotheses, we always begin by assuming the null hypothesis is true The logic of hypothesis testing is such that we compare empirical reality (i.e., our sample) to a standard of what we would expect if no relationship or difference between groups of cases existed in reality  Hypothesis Testing Cont’d Hypothesis Testing follows a five-step model: 1.Make assumptions and meet test requirements 2.State the null hypothesis 3.Select the sampling distribution and establish the critical region (i.e., set alpha) 4.Compute the test (obtained) statistic 5.Make a decision and interpret the results of the test  Hypothesis Testing Cont’d When you set your critical region (step 3) by establishing the alpha level, you are cordoning off the area beyond your critical score Remember, alpha is associated with a critical score – either a Z, t, or Chi Square score (depending on the type of test) if you have a large enough sample size, and you are conducting a two tailed test, you know that you’re splitting the area beyond the critical score into each tail Consequently, with an alpha of 0.05, you will have a corresponding critical score of 1.96 You’ll have the same t score if your sample size is large enough  Hypothesis Testing Cont’d If, in step 5, if your obtained score is larger than your critical score, you can reject the null hypothesis Or, if your p value is smaller than your alpha, you can also reject the null hypothesis To lower your chances of committing a Type I error, you set alpha at smaller Because both represent statistically levels, because it will be harder to get an obtained Z, t, or chi square score that is significant results greater than your critical score Likewise, it will be harder to get a p value lower than your alpha…  Now, we looked at a few types of hypothesis tests… 1. Chi Square 2. T tests (one-sample and two-sample) Hypothesis 3. Pearson’s Correlation Testing Study THESE!  Chi Square Chi Square is the most frequently used hypothesis test… It is most appropriate for nominal and ordinal level data To compute Chi Square, we use bivariate tables There are two dimensions to bivariate tables (horizontal and vertical) We refer to the horizontal dimension as Rows (the DV goes here) We refer to the vertical dimension as Columns (the IV goes here) The intersection of the rows and columns are called Cells Each column or row represents a score on a variable, and the cells represent the various combined scores on each variable  Chi Square Example Research Question: is membership in volunteer associations affected by place of birth?” This gives us two variables: (1) “Place of Birth” (which is the IV) (2) “Level of Involvement in Volunteer Associations” (which is the DV) Furthermore, let’s say we sampled 100 people If the variables are independent of one another, it means that the cell frequencies are determined by random chance alone This would mean that about half of the respondents The pattern you see here indicates no born in Canada rank high on participation, while the other half of Canadians rank low relationship between variables (they do And the same pattern would present itself for people not affect each other) not born in Canada  Chi Square What Chi Square does it test the expected frequencies of what would occur if no relationship existed against your actual observations This is to say, “the greater the differences between expected () and observed () frequencies, the less likely variables are independent and the more likely we will be able to reject the null hypothesis” (Healey, Donoghue, and Prus 2023, 222-223). When we use the Chi Square test, it produces a test statistic The larger this test statistic, the greater the association between variables – meaning the more likely there is a relationship between variables and the more likely that relationship is to exist in the population If you have a large chi square value and the p-value associated with your chi square is smaller than your alpha – again typically set at 0.05 – then you can say that the relationship exists in the population…  Interpreting a Bivariate Table But the Chi Square only tells us if there is association between variables, it does not tell us if the association is positive or negative To do that, you need to interpret the bivariate tables… You need to examine cell percentages For example, let’s say we were interpreting the relationship between the level at which a country has transparent laws with predictable enforcement and the level of executive corruption in countries… Interpreting a Bivariate Table Cont’d Look at the percentages across rows You can see here, executive corruption is markedly higher in countries with no enforcement (84.8%) compared to countries with high enforcement (12.5%) Conversely, in only 3% of countries where there is “no enforcement” is corruption “rare”, compared to 58.8% of countries where there is “high enforcement” Therefore, as enforcement increases, corruption  Interpreting Chi Square and p Value Likewise, the p value is lower than 0.05 (see Asymptotic Significance (2-sided)) Moreover, a chi square value of 61.571 is significantly larger than the critical value of 7.815 – which is the critical value for a Chi Square test with 4 degrees of freedom Therefore, there is a relationship, and it likely exists in the population  Measures of Association We also looked at a number of measures of association… Measures of association provide us information on the strength of association between variables When applicable, they also tell us about the direction of the association – i.e., positive or negative Two variables are associated with the distribution of one score changes as the result of another  Measures of Association for Nominal Level Variables Here, we looked at two measures of association: 1. Phi 2. Cramer’s V Both Phi and Cramer’s V range from 0.00 (no association) to 1.00 (prefect association) The closer they are to 1.00, the stronger the association; the closer it is to 0.00, the weaker the relationship. We use Cramer’s V when the bivariate table is larger than 2x2, because if the table is larger than that Phi can exceed 1.00  Measures of Association for Ordinal Level Variables We looked at four measures of association These are calculated using the logic of pairs for ordinal level variables: (i.e., the various rankings of respondents on 1. Gamma (G) the X and Y variables) 2. Somers’ d () 3. Kendall’s tau-b () Gamma is the only measure not to take tied pairs into account 4. Kendall’s tau-c () Therefore, if may overinflate the strength of association between two variables  Measures of Association for Ordinal Level Variables Cont’d When we have an unequal number of What separates Somers’ d is that it will categories on each variable, we use provide you a different score depending tau-c instead of tau-b on which variable you use as your independent or dependent variable For example, if you have a bivariate table with 3x3 or 4x4, you use tau-b As such, it provides you different predictive power between two variables For example, if you have a table that is depending on which you choose as X 3x2 or 4x3, you use tau-c and which you choose as Y  Measures of Association for Ordinal Level Variables Cont’d Because these measures are used for ordinal level variables, they can also tell us about the direction of the relationship They range from -1.00 (a perfectly negative relationship) to +1.00 (a perfectly positive relationship) If the value is negative – it is a negative relationship If the value is positive – it is a positive relationship  Cohen’s d Another measure of association (technically a measure of effect size) is Cohen’s d It is used when you conduct a two-sample t test (because you use the means of two independent samples to see if there is a significant difference between groups)  Pearson’s r The last measure of association we looked at was Pearson’s r It is a measure of association between two variables measured at the interval-ratio level It ranges from -1.00 (a perfectly negative relationship) to +1.00 (a perfectly positive relationship) It is interpreted in the same way that measures of association are for ordinal level variables