Fordbid Title Field with airSlate SignNow

Fordbid title field

this is the first lecture where we go into a bit of detail about linear models now I've already actually talked about linear models quite quite well at least in one lecture I was mentioning it a bit but this is the first lecture where we start to look at it in detail so hopefully some of the things we cover will seem familiar from the first week of term and we'll just sort of push it push it forward a bit so what we're going to cover today I'm going to talk about what what the linear model is describe how we assess the fit of models and I'm also going to talk about some issues around how we estimate parameter I've talked about this before estimating parameter so we get these beta values and I'm going to talk a bit about some things we can do to make sure that those betas and significance tests around these betas are accurate and in your practical class you're going to start putting this all into into practice using SPSS and they'll they'll talk a bit more about interpreting what comes out of SPSS as well okay so we've seen before what a linear model is and it looks like four points or today it's green nice we've talked a bit about what linear models are so I Chuck I've tucked up this equation before and as sort of a nicer way to put this equation is to replace some of the some of the symbols with some words that actually make sense so we're trying to predict an outcome which we normally denote by Y from some kind of model so before when I put that up I just would have put the word model to try and keep things really simple but we've seen that for example the word model can be replaced with this thing and this is a linear model or the equation of a straight line and whenever we make predictions for a model there'll always be some kind of error attached to that prediction so it will never be perfect so what we're talking about today really is the bit more of the pragmatics of fitting this kind of linear model and also looking at how this model extends to more sort of slightly more complicated scenarios so when we looked at this before the equation of a straight line we had a bit of a chat about these kind of beta values and what they represent something's not a bit weird with my screen half my slides missing and does it do that okay so it's not it's not a personal thing it's not a vendetta that machine has against me okay well we'll muddle through as best we can I'll just have to point at things that don't exist and I hope been using so the beat of one in that equation is well actually both the pieces are known as parameters and that's just a kind of a confusing name for I don't even know what for it's a confusing name for a parameter a parameter is just that it's just the B right so these B's represent certain things so we've seen in week one we had a talked about this before and the the beta one so a beta associated with a predictor variable represents the strength or direct or and direction of the relationship between that predictor variable and the thing that they're trying to predict so that visa tells us something useful about the relationship between the predictor variable that's the X in the equation and the outcome variable there's also a beta zero in there and that tells us something about the value of the outcome variable if all our predictors were zero so in the simple simple scenario where we've only got one predictor this is often referred to as the intercept and you're seen it graphically is you know the point where that if you've got a regression line the point where the line crosses the y axis but I want you to think about it more in terms of it's the value of the outcome that you would get if all the predictors were 0 so in a way you can think of in there in the simple case where you've only got one predictor you can kind of think about both of these things as defining the shape of the model and as you'll see as we extend the model to have more predictors that still applies the beast is where you tell us what the shape of the model is so they really define is kind of what it looks like basic so I kind of equate this a bit to if you've ever been to West Street which I've been burned into a knife for a very long time indeed but West streets like where all the nightclubs in Brighton are and if you go there at certain times of day during the week there's often lots of drunk people staggering around and when you're drunk you kind of come out the nightclub you don't really know where you are or what you're doing and that's a bit like a regression line left to its own device is a regression model or a linear model will just float around in mathematical space without a caring world okay I know that I all I'm doing now what the b20 does is that's like a kind of it's a it's a kind of a it's a kindly policeman if you write he doesn't arrest the regression line but instead he tells it where it is it says you're in West Street it may even nail his feet to the floor or something so it positions it in geometric or mathematical space it gives it a place to stand if you like so long ago an aggression line it goes right there you are you are fixed there and what the b21 tells us is it tells us in a way how drunk the regression line is because it tells us the shape so is it leaning this way or is it leaning that way is it what it actually would never be so completely vertical or is it like completely like horrid or whatever I do this I hurt myself is it completely how is it completely horizontal now for regression lines horizontal I was right I hurt myself again if it's completely horizontal what does that mean it means there's no relationship at all with the regression line is flat if the beta is zero that means no effect whatsoever so it's like you know it's got so drunk it's just dead no effect at all so the beaters tell us something about the shape of the model so we can start using some slightly frivolous but real examples of what we might predict and from what so we could predict geekiness from example for example from how much you like statistics so if you had some kind of psychological index of geekiness that may well be predicted by your love of statistics because you know people think people think that people like stats also like other things like sci-fi it's not true at all so that's that's an example of a simple model and we're going to see how we can extend it now the important thing is we've got this b20 it tells us how geeky our sample would be if if their love of statistics was zero and we've got this beta one which tells us about the relationship between these two variables so if this beta 1 is positive it means that loving statistics makes you boil or doesn't necessarily make you but is related to being more geeky if it's negative it means that loving statistics is related to being less geeky and if it was zero it would mean no relationship at all between loving statistics and being geeky so just around this home beaters tell us about the shape of the model so they define the model for us they tell us what it looks like so for example we can look at three models that have the same dear oh dear oh dear that have the same intercepts so they have the same value of the outcome as each other when the predictor is zero but they have different shapes so the B's are ones are different now this is an example of your verbal coherence as a function of how much you drink so if you're a really hardened drinker you'd have a beater of zero so the more you drink so drink a quantity of alcohols going going up along this axis and so the more you drink it has no effect whatsoever on your verbal coherence you start off as verbally coherent as you are which is sort of midpoint of the scale and that just doesn't change you drink and drink and drink and drink drink nothing happens that would be slightly unusual what normally happens so for most people in the population we get a negative relationship so this would be a negative beta the more you drink the less coherent you become so you know a certain point can probably just incomprehensible for everyone and there it's not inconceivable that you could have a positive relationship perhaps if you bit socially anxious or what I'm saying normally quite shy so maybe you're not particularly verbally coherent when you're sober because you get very anxious and worked up but a bit of alcohol suddenly kind of loosens the corpse if you like and you become slightly more verbally coherent cuz you've stopped worrying about you know what you're saying things like that so this just shows you how the beat is define the shape of the models that we fit in a very simple scenario the opposite situation would be if we have a look at models that add the same shape so they will slow down so this is as I said this or typical scenario the more you drink the less coherent you become so they're the models also pinned down but they have different intercepts so the samples have different natural levels of verbal coherence at that's all zero basically so when when they're sober so this would be a sample of in people are very very coherent and these people are less coherent to begin with I just want to make the point that these beaters whatever the model looks like however complicated it gets they just tell us about the shape of the model that we're fitting so they define what it looks like so let's look at an example this is example it's gonna be used in your handout with the example from the book and this is an example of predicting record sales or I think you know in a fit of trying to keep to the times I may have updated this to album sales you know just people don't buy record whereas some of them do I still buy records but I'm old and so anyway album sales or whatever you want to call them I know downloads whatever the kids are into these days and so this is like thousands of record sales and we're trying to predict it from advertising budget because how the record companies or any company try to sell things they spend money on advertising so a very simple model might be to just fit a basic one predictor kind of kind of model so can we predict record sales from the amount of advertising that we spend the answer to that is kind of yes so we've got a nice nice kind of linear trend there and it's pointing upwards so more you spend more records you sell what happens if you want to expand this model so if we're not just content to look at advertising but if we suddenly decide well you know advertising is all very well but we can probably do a better job of predicting record sales if we measure some other things that might be important so for example we might measure airplay on radio because that's very important people listen to the radio they hear songs before they know if they like the songs and then they want to go and buy the album this is a you know it's kind of like the mere-exposure effect really in real life so you get this a lot before records come out you'll hear them on the radio lots and lots and lots of times until yeah even there are there are their records well so for example when I was when I wrote lad and I used to hate a lot of I was sort growing up in there Isis well a bit of a 70s for something maybe the 80s and you know so I used that as you might imagine like that you know I am made and all stuff like that but I wasn't particularly popular and all the stuff in the charts was kind of things like wham and rhymes anyone I can think of now actually for anyway so there's a lot of Wham in the charts and I hated it with an absolute passion because you know like when you're a teenager you kind of have a a Bible for these sort of things that you know it's quite phenomenal so you know I'd be listening to my Motorhead albums or whatever and if I heard wham or heaven forbid I actually went to like a school disco or something we used to have them back then and at least occult discos probably have like school nightclubs now um and there was some Wham on I just oh wow I hate wow hey but over like the last 20 years you know where you keep hearing this stuff over and over again now when I hear Wham I mean I don't want to go out and buy it buy it sort of thing yeah it's not so bad don't know what I used to get so riled up about so you get this mere-exposure effect so the man you play stuff on radio or the man you hear stuff you tend to not always the case but you tend to end up liking it more my parents for example became pseudo Iron Maiden fans as a result of listening to what was coming out of my bedroom for five years anyway so if you want to add another predictor how does that model change so it's very easy to conceptualize one predictor we have a line it's all nice and everything if we add predictors we get something called a regression plane that's not the crappiest joke you're here on this course but it's one of them so you get a regression plane now what does the regression plane look like half of it looks like this so rather than having a sort of nice graph with two axes you end up having an axis in sort of third dimension and your plane sorry your line changes from being a line to being a plane so this is a nice little video it's not the most entertaining video it has to be said but it's just to kind of illustrate the shape of the plane that we would have fitting this model so one of our axes is now representing plays on radio one of our axes is advertising budget the yellow dots are the data that we've collected and the green thing is the regression model so it's it's a line that turned into us or three-dimensional plane for the basic principles that are applied with a line still apply it with this plane so the basis that we have as we'll see in a minute still define its shape the intercept still locates it in some mathematical space for want of a better phrase everything still applies it's just where we're kind of we're making things more complicated and the model starts to look more complicated but essentially all the principles are the same whether we have one predictor or many predictors so what does this look like as an equation if you add predictors to a linear model people typically refer to this as multiple regression the multiple referring to the fact you have multiple predictors not the fact that you're doing lots of regressions each predictor will have a parameter associated with it so that's a beta value that's kind of latched on to that specific predictor so we have one predictor our model hopefully you might have a handout model looks like this thing that would be down here if it is projected and so you've got the b20 you've got one predictor an X and a beater associated with it if we add another predictor in or more predictors in we just add them in so we added a new X and you know we could give that a better name than X we could call it airplay or something like that and again we give it a beta which tells us something about its relationship to the thing that we're trying to predict a Nordic waiting at the bottom for those of you that can see it on your handouts tells us is the dot just means that we can carry on adding things in until we've run out of predictors that we want to include so the linear model one of the beautiful things about it is it's in a way it's very there's a very simple idea is that you put a variable in and although in these equations when you tend to denote them by X's which is not here not the nicest way to denote them but you can think of those X's is just the name of the variable we could just plug them in and whenever we plunk one in we have to give it a beta we will give it a parameter that estimates what the relationship between that predictor and the outcome is in the population so bases are touched or not I've said this like a million times but you can never rewrite it is too much so the base is attached to variables tell us something useful about the relationship between a particular predictor and the thing that we're trying to predict so few examples of multiple regression or regression with several predictors we could look at how bored people are this plays on my mind a lot based on how much time they'd spent at a lecture for example so this could be a variable ranging from 0 to 50 minutes presumably you know your your boredom goes up the more you're listening to me drone on but we might also add a different predictor so this is a different X but I've given it more useful name of the topic that I'm talking about perhaps you'd be less bored if I was talking about what I did at the weekend rather than talking about linear models it's questionable whether that's true but it could be the case now the interesting thing is that would that would be a categorical variable am I talking about stats versus am I talking about what you know Jolley activities I've got up to as a weekend and you might think well surely you can't put that in the linear model actually you can in a couple of weeks time we're going to look at that in more detail I just want to flag it here that you can have categorical predictors you just have to include them in a in a very specific sort of way another example hangovers how bad is your hangover well obviously that should be a function of how much alcohol you've had to drink the night before more you drink worse your hangover should be but we might also factor in another variable such as how much sleep you've had I don't know arguably the less sleep you've had the worse your hangover will be I don't know that's possibly not always the case I don't know um and we could even extend this further to add another predictor so perhaps personally I don't smoke but I know some people that do and some of them have said to me that they think you know if they've really had a lot to smoke as well as a lot to drink their hangovers are worse so maybe might want to factor that in as well so we could change our model to just include another predictor variable of how much how many cigarettes they smoked or whatever else they might make um and you can see the models just extending each time we're just literally putting in a new predictor variable rather than using X's I'm just giving them a bit more helpful names like the name of the variable but the point is each one has a beta and this beta is going to tell us about the relationship between a particular predictor and the outcome so in a way it's a very simple idea you've got as I said one equation the only equation you ever really need which is this equation of a straight line and we just saw tweak it a bit we just adding new predictors and each time we do we give it a parameter that we try to estimate so the idea behind the linear model is really quite straightforward now when we have multiple predictors as I said we get a regression plane so this is going back to our others is going back to Doctor Who it's going back to our example of record sales based on the advertising budget and number of times a record was played on radio and all I want to point out is that the beaters as well as telling us about the relationship between a predictor and an outcome variable they also do define the shape of this um FCR define the shape of this plane so the beta zero will be the little little peak whether the planes or hits the y-axis or is the is the value of the outcome when both predictors are zero the beat are associated with radio play will represent the slope of this side of the regression plane so this would be the beater associated with the radio play and the beta associated with advertising budget would represent the slope of that side of the regression plane so basically those beaters tell us where the regression plane is located and also what shape it is so you imagine business I know it's a crap prop um I'll admit it imagine this is your regression plane so it could be completely flat which would be no relationship at all between either predictor in the outcome or it could be sloped like that could be sloped like that it could be sloped like that so it's like it's flying around it's flying around geometric space the beta zero will lock it in to a certain place that's a right you are there and then the beta ones and beta to use tell us about you know how it's whether it's at that angle or that angle or that angle or whatever so the beaters literally define geometrically what the model looks like but in more practical terms they tell us about the strength and direction of relationship between predictors and outcomes so nothing really changes now if you had a third predictor this is where the TARDIS comes in it gets more complicated to visualize the model because you'd have to go into like the fourth dimension and you have like five or six predictors you'd be going to the fifth and sixth dimensions and everything gets very confusing I'm not really sure how you visualize that maybe like in regression playing that and send time and space or something I don't know but anyway the basic principles apply you can just keep adding in predictors and the beaters will tell you about the relationship between those predictors and the outcome but also if you could visualize you know in five or six dimensions what what the regression model looks like those pieces would also be defining its shape so the betas are pretty important so I'm going to talk a bit about them now we've talked about parameters before in the first week I believe in the context of the linear model these beaters are as I said they're estimates they're known as parameter estimates so the parameters are the actual values in the population they're the things that we would like to get hold of but we can't so instead you know we fit a sample we fit a model to the sample data that we've collected and using that sample we estimate the parameters so they're called parameter estimates because they're not we can't get the actual values in the population so we're estimating them based on our sample now we also came across the idea in week one that any parameter at all whether it's a mean or whether it's a regression coefficient I were talking about here you can construct a confidence interval around it and that conference interval will give you the the boundaries within which the population value is likely to fall now remember these beaters represent the relationship between a predictor and an outcome and also remember that if a beater is zero if the line is dead flat that means there's no effect at all if a confidence interval contains zero so if zero falls within the boundaries of it what that means is it's possible that the actual effect in the population of the value of the parameter in the population could be zero in other words no relationship at all so if you see conferencing tools that contain zero that means that basically the predictor could have no relationship whatsoever with the outcome so in other words you know it's not very good predictor you can also more formally test predictors using a significance test and actually it uses a t-test and what that test does is to compare the value of the beta that you've estimated against zero because if you think about zeros are sort of null hypothesis like I said zero is a flatline no effect whatsoever so a significance test of whether beta is zero is going to tell us whether or not the the predictor is significant so if you get us over p-value less than point and five it means that that predictor is significant but bear in mind what I said before about significance and large samples and things like that it's not always that useful what is definitely useful is the fact that the beta also in its own right is an effect size and I talked about effect sizes in week one and I said they're useful because they're not so they're not so biased by the the size of the sample that you have so they you can interpret them independent of significance and things like that so it may be for example that you know you fit a linear model in quite a small sample you get a you know a big substantial value of beta that looks like there's a sort of an impressive relation and shift between the predictor and outcome but when you look at the significance it's not significant just because you've got a small sample so all of these things are things when interpreting beaters they're things to consider what is the confidence interval look like what's the significance test and what what is its value in its own right irrespective of everything else we also saw a weak one the parameters can be biased by various things so we talked a bit about very broadly in the second lecture about how we estimate parameters and I gave a very crude description of what's known as the method of least squares now in regression we still use the method of least squares but you'll sometimes see it referred to as OLS which stands for ordinary least squares but it's the same principle that I was talking about in week two so the idea that we estimate parameters that give us the least amount of squared error we also saw in the maybe the third lecture I think when we were talking about bias that if we do significance tests or if we construct confidence intervals around parameters they depend upon certain things being true one of them is that the sampling distribution of that beans should be normal we should have homoscedasticity or homogeneity variants whatever you want to call it and our observations should be independent from each other so we shouldn't have related observations if those three things are true then we can trust the significance test and confidence intervals that are generated we also saw that our parameter estimates that the estimates of beta are optin all using ordinary least-squares when the errors in the model of normal and again if we have homoscedasticity so I'm going to talk about this a bit more next week what I want to focus on now is this idea of how to ensure accuracy of the significance tests and confidence intervals and we can use this thing called bootstrapping so next week we're going to look at how we identify you know non normal or non normality and heteroscedasticity and we've already talked about it a bit in Week sorry lecture three if those assumptions are violated so if we think all we can't trust the significance test we can't trust confidence intervals we can do something known as beach trapping which is fine now going to attempt to explain and so right the idea behind the bootstrap we've got this sample of scores of thirty scores that we've got and what we're trying to do is to use this sample to create what are known as robust estimates of the confidence interval and the actual parameter itself so the bootstrap is a process through which we can generate robust or so-called robust estimates now the bootstrap works on the principle of using the sample data to generate robust estimates I'm really reckon turning that off by the way and so how does it do it right well I need my pink hippo because whenever I'm explaining anything complicated the pink hippo needs to come out and he's living in a box today so imagine we have our twenty five schools actually ping-pong balls but each ping-pong ball has a number on and that number relates to one of the numbers that was on the frequency distribution that was up there a minute ago so got 25 scores all represented on a ball so this box basically contains our sample of scores now let's say we wanted to estimate the mean of these scores or let's say we wanted to do a bootstrap mean so it's just a bit simpler to explain so the parameter where entry fram's that we're interested in is the mean so what do we do well first of all we need to construct what are known as bootstrap samples so this is something that SPSS does and you don't really need to worry about it doing it because it does it very quickly but what it's doing is creating these bootstrap samples so what's a bootstrap sample well imagine you've got your sample data and imagine your SPSS you pick a value out just one since the value of three so this is your first value and then you give it to the magic pink hippo that lives inside SPSS and he memorizes that value so he memorized that value of three and it looks like he has a small brain but in fact it's enormous so he memorizes it and then he spits it back into the song and then he takes another one it's a one remember Issac it's quite important that he eats it to memorize it or what you know sucks on it for a bit like a Gobstopper and so he he memorizes it again so he's got a three and his brain is now at one his brain and then excuse it back in so this is known as resampling because we're taking a sample score and then we're putting it back should I pick on someone else then we take another number so this one is a zero again hippo memorizes a zero he's got free he's got a one he's got zero in his brains game thinking about it thinking about it memorizing it spews it back in though he takes another I need to I need to do this for a while while this kind of warms up so he's got to now I'm memorizing the T memorizing memorizing the t he speeds it back in so he does there's lots and lots of time so he's randomly sampling a number he's memorizing that number and then he's putting it back so he could he could pick the same ball twice and that's fine it doesn't it doesn't matter so he's putting it back each time and that's known as a bootstrap sample so that sample could be you know it could be five scores or it could be as big as the sample itself because each time we're putting back in the score so what he does then is it's got a bootstrap sample it's all in his head and then he goes and he works out though it works out the parameter within that sample so here we go okay you've got a bootstrap sample what's the mean of those 40 good 2.4 apparently so we don't have what's known as as well it's a parameter from the bootstrap sample so we've got one parameter one bootstrap sample and then because he really likes doing this he does it again so go pick the ball memorizes it it's a two picks another ball slight Hungry Hippos quite literally he memorizes a number takes another one Oh another one always says that one doesn't taste very much so he does that again that's another bootstrap sample he's taken say you know ten numbers out he's memorized them all and then works out the parameter it's one point six so in his previous bootstrap sample he got a mean of two point four this time he's got a mean one point six oh my god he's gonna do it again 2.23 bootstrap samples now each time he's randomly taking numbers out of the sample he's worked out what the parameter he's interested in for me and he's kind of made a note of it somewhere on a little notepad somewhere on the screen in fact guess what he's gonna do again this is very tired get more frantic but he's done it all the same process his brain is explaining now he's never got for bootstrap samples he's that calculated the parameter in each one now it's 4.2 so you can see the parameter estimates are different in each sample and that's fine because we're not interested in them individually what we're interested in is doing this lots and lots and lots of times now you'll be glad to know I'm not going to demonstrate this for the two thousand times that SPSS will do it but it's this basic principle of creating bootstrap samples working out the parameter and then SPSS will do that a thousand times so we'll end up with a thousand bootstrap samples the hippo of bootstrapping will be very tired by the thousand some could be right but what we'll have is a thousand bootstrap samples and a thousand estimates of the parameter now I decided to not have a thousand numbers on the screen okay fact of Sussex decided to not let me have more than ten on the screen but anyway so the principle is we've got lots and lots and lots of these numbers there's another three rows of these numbers below now what's interesting about that is we could then take say we want to know what the what they say the bootstrap will robust mean of these samples are well we've got say a thousand means a thousand estimates based on a thousand bootstrap samples you can just simply average those estimates and that will give us a kind of a boot strapped average if you like so that's one thing we can do we can use bootstrapping to actually estimate the parameter in a in a way that's unaffected by the distribution of scores by taking samples from our sample these bootstrap samples and working out the parameter within each sample and then averaging across them what happens more often than doing that is we rearrange them in order of well again SPSS does this behind the scenes you can rearrange them in order of size so these are just going to disappear and then hopefully a materialized in order of their size again this will be better as you can see the whole screen but essentially there's 40 scores here and then now they're the same scores as before it's just I've ordered them from smallest that was the smallest bootstrap parameter that we got through to the largest bootstrap parameter which on my screen is 7.6 now having arranged them in order we can literally compute a confidence interval based on the limits between which 95% of those values fall so in this sample of 40 bootstrap samples that would actually involve just cutting off the top and bottom score so this is the point at which the middle 95% of scores lie so when SPSS is creating a bootstrap estimate of the confidence interval this is sort of what this is known as a percentile bootstrap this is what it's doing it's taking like a thousand bootstrap samples it's working out the parameter in each bootstrap sample then it orders them and then it works out the limits between which 95% of them four so rather than normally when you work out confidence interval you do it sort of using equations and this is sort of the way of doing it empirically you sample from your sample and you work out the parameter in each of your bootstrap samples and then you work out the limits within which 95% of them fall I just realized I haven't needed a hippo on my hand for about the last three minutes but it's reassuring comforting so this is what bootstrapping is it's taking these bootstrap samples it's working out the parameter you're interested in within each sample and then you can construct a confidence interval based on the limits within which 95% of those scores fall now from a practical point of view you don't need to worry about all that going on I just wanted to explain what's happening in the computer as far as you're concerned you will just get if you opt for bootstrapping on SPSS you'll get to two output table rather than one so you get a normal output like this don't necessarily make any sense to you into each on your practical class but essentially we've got predictors we've got beaters so estimates of what the beaters are and then estimates of their significance and also the confidence interval for each beater so for example the population value of fur this essentially so the relationship between number plays of radio and how many record sales you've got the sore point estimate in our sample is three point three seven the confidence interval is two point eight two and three point nine two so basically the population value is likely to fall between those two values now what you guys are table underneath which has bootstrap versions of those significance and confidence intervals so all it really means is if your assumptions are broken and just ignore these confidence intervals and these significance values and look at the ones down here instead so from a practical point of view if all the restarting stuff didn't make any sense to you at all basically you just look at this table and ignore this table essentially so that's a good practical way if you've broken assumptions of just knowing that your confidence in terms of significance tests will be what's known as robust it means they're unaffected by the fact that your distribution is is not normal or you've got heteroscedasticity so having a estimated these beaters and with so we fitted a model we kind of know it looks like we have to then try to assess how it fits and this there's sort two ways of doing this which are commonly used so to change example a bit I'm going to have a look at whether using Facebook turns you into a monkey because there's evidence that it does this is a bit anecdotal I know this is a good friend of mine who got really obsessed with Facebook a couple years ago like you can do anything until you've been on Facebook for a few hours galloping more than gotten Facebook before going to work now it works he was on Facebook a lot of the time and then eventually she's got the sack because he was on Facebook so much and eventually she ended up living in a gutter very sad story but also she turned into a monkey so I thought you know I doubt whether the amount of time you spent on Facebook actually can turn you into a monkey so I did a bit of study I measured people's monkey nice I mean before Facebook was in meds if my arms were completely unheard but look at the state of them now it's only gonna get worse with Twitter so we got monkey this it valuated on a seven-point scale and down here would be ours on Facebook so there's a few things we can assess these are the dates that we leave that we've collected now remember that if there's no effect whatsoever if there's no relationship between these two things you'd get a flatline so we can represent this by the mean of all the scores that we've collected completely flatline it predicts no matter how much time we spend on Facebook it predicts the same level of monkey nests which is four on the scale now if you remember back to the first week we we have a look at how we can compute error in or look at fit and we said we could use deviations so we could use the difference between what our model predicts so that's the line and what out at the actual scores we observed so this is when we look at things like variance and standard deviation so we had these deviations or also known as residuals or errors a all the same thing and if you remember what we said was we can't just add these up because some of them will be above the line and some of the below the line so they'll kind of cancel each other out so if we want a total it's no good to just simply add these deviations because some will be negative some will be positive so what we do is we Square and then we add up the squared values so this is what we're doing here we can work out a sum of squared error for a sort of a baseline model a model where there's no effect at all and that is the overall mean of the of the 19's and this will give us an idea of how much variability there naturally is in monkey nests if we don't include our predictor as a predictor so this is just telling us how much variability in scores we up monkey nough scores we have in the first place so if there's no effective if we fitted like a flat model if you like how much variability in scores do we get we can then fit our model so the model we fit will look like this so it's a straight line and again we can look at these deviations so our model will for this score it predicts the point of where the line is so predictive value about 5 but actually their monkey news scores a bit higher here we would predict a score of 3 in a bit but actually their monkey news scores a lot lower so it's again we've got these little errors in prediction so we can work out the error in the model in exactly the same way so it's just using this idea of sums of squares so we work out the deviations the difference between what's predicted by the model and what we actually observe and we square them and then we add the now hopefully if our model is good the error that we get when we fit the model will be a lot smaller than the error we get when we just use the mean of scores when we use a line representing no effect at all and that's known as the model sum of squares and it's actually it's the difference between the two lines so you need to kind of sort of ignore the data points of it here so it's saying for this data point what's the difference between what the mean predicts and what the line predicts for this data point here what's the difference between what the mean predicts and what the line predicts and this is actually it's the difference between the other two sums of squares that we talks about but this represents improvement essentially saying you know if we use the mean this is this is the prediction we get we use the line this is the prediction we get what's the difference between the two now if we have several predictors the same principle applies is just rather than rather than our baseline being a flatline our baseline becomes a flat plane so this is all this slide is there to show you really that the principle I've just talked about it applies if you've got lots of predictors or if you just got why it's easy to explain if we if we look at one predictor but if we had another one we'd get exactly the same scenario it's just our baseline model would be a flat plane rather than a flatline so to test the fit of the model we can use sums of squares and I told you earlier on in the module that sums of squares are important we can use total sums of squares which tell us this or the total variability in score so you can imagine yet if you'd like baked a cake this would be how big your cake is to begin with we can look at the error in the model so we can look at the residual sum of squares which is the variability or the stuff we can't explain the errors leftover when we fit the model that we fit and we can look at the model variability the model sum of squares which is the difference improvement if you like that results from fitting the model compared to not fitting it so like I said I like my cake analogy so just to demonstrate it a bit more visually the types of summer squares is is like this this big round cake so this is all the all the variability among keenest that ideally we would like to be able to explain by adding predictors to the model so we had a predicted to all or several predictors to the model and then we work out how much that improves it and we assess that using the some of squares and that will give us a slice of cake now hopefully a big slice of cake because we want we want to be able to explain a lot this so we're hoping that our model some squares is a massive slice of pie but there's still going to be some error left over so that will be you know hopefully a small slice of pie and relatively speaking now this this model isn't strictly accurate because we can this should literally dissect into like two halves or not very hard for to two sections not three but it's the only cake a picture that I could find so we're working out how much variability there is and we're kind of slicing it up into two bits and then we use these two bits in one of two ways the first is we can work out what's known as an F ratio and the F ratio is basically it's it's we end up using I'm going to come into this a bit more in a few weeks time but we end up using something called mean squares which is like the average sum of squares rather than the sum of squares themselves but the what they represent is the same thing so don't worry too much about that you can solve conceptually think of them as sums of squares so the F represents the improvement due to the model so it's the model sum of squares or mean squares divided by the error in the model so the F ratio is the ratio of how much the models improved relative to how much error there still is left over so obviously if that ratio is big if the F value is big what it means is that we're explaining more variability that the model is is improving our prediction a lot more than it has error in it if you see what I mean so it's kind of it it's it's always the ratio of how good the model is compared to how bad it is so if that ratio is big if the F value is big it means the model is a lot better than it is bad so it's more good than bad the other thing we can do is to use something called R Square and this is just the it's the ratio of the improvement due to the model so the model sum of squares relative to how much variability there was to begin with so it's literally the proportion of variability that is explained by fitting the model so you can I mean SPSS computes this you know using these sums of squares it also if you if you look at the values of the outcome that you would get from using the model so the predicted values that's what they're known as and this is this R square is actually that it's kind of related to the correlation between the values you actually observe and the values that the model predicts so it's literally the Pearson correlation between those two things squared the other thing worth mentioning so the r-square is useful because it tells you proportionately how much variance is explained so you can come up with uh solve you know we fitted our model it explains 30 percent of the variance or 10% of the variance which is quite a useful measure in a way the other thing to be aware of is SPSS throws out an adjusted version of this which you'll see in the output when we do a practical class and what this does is to estimate the value of our square in the population so the R square that you normally look at is based on your sample data but the adjusted r-square is an estimate of what a square will be in the population so it tends to be smaller so your x gets referred to as a measure of shrinkage so it's a measure of how much basically how representative the a square you have in your sample is of the population if the values the Artic only just a square very similar then it means that your R square is a is a sort of good approximation of what's going on the population so your sard what is therefore presumably a good approximation if the adjusted r-square is a lot smaller than what it means is you know your sample data is quite different from from what you're likely to get if you'd been able to test the whole population so they're kind of the key concepts to take into your practical class first is the idea we can fit linear models we can extend them to add multiple predictors each time we do we have to estimate a beta the beetus tell us about the shape of the model the size and the direction the relationship between the prediction the outcome we can significance test them so we can test whether they're different from zero we can intervals around them for those significance tests and confidence intervals to be accurate we need to assume that the sampling distribution of them is normal and we can use bootstrapping if we're in doubt about that to give us a pretty accurate estimates and finally we assess the model generally how well it fits the data using either an S test which is sometimes known as ANOVA or this R square statistic so all of this is going to you're going to do some examples in your practical class and you know come up against these concepts again and hopefully yeah hopefully that will kind of be and drill them in a bit more and then next week we're going to look at and working out whether the model we have is kind of biased or not biased and a good representation the population