Mini-Troughput: The Triple Lindy Stats Flop

Avatar

Michael Siegel

Michael Siegel is an astronomer living in Pennsylvania. He is on Twitter, blogs at his own site, and has written a novel.

Related Post Roulette

59 Responses

  1. Avatar Pinky
    Ignored
    says:

    I’m confused. Doesn’t the original tweet say the IHME projections fit well? There are four projections on the graph, and the red one is the only one not labelled “IHME”. Are all the IHME projections cubic? If not, why are people making fun of the cubic curve?Report

    • Avatar Michael Siegel in reply to Pinky
      Ignored
      says:

      The cubic one is now the one they are favoring over the IHME (which is more complex and does not yet account for loosing restrictions). I’ve edited the post a bit to make this clearer.Report

      • Avatar Pinky in reply to Michael Siegel
        Ignored
        says:

        OK, but it does contradict Tuesday’s post from the CEA, which says “As shown, IHME’s mortality curves have matched the data fairly well.” That post is featured in your article. Kareem El-Badry’s tweet is a reply to it, but that post specifically endorses the other three methods shown on the graph. Are you sure you’re reading this story right? The CEA sends out a post with four curves and mentions three of them, and now people are complaining about the fourth?

        And it isn’t like “the curve” is some radical, simplistic departure from the way we’ve been talking about the pandemic. We’ve been bending and flattening that bad boy all year. So why are people suddenly complaining about it?Report

        • Avatar Michael Siegel in reply to Pinky
          Ignored
          says:

          I didn’t want to go too much in depth here, but the White house has been vigorously defending the IHME model, which predicts way fewer deaths than anyone else. The tweet was meant to defend that. But it also included the “cubic model fit”. And we found out that they are basing policy not even on the uber-optimistic model but on that cubic model fit. So the conversation is like so:

          “The IHME predicts way fewer deaths!”
          “Their models are too optimistic.”
          “Well here’s a graph showing it’s done a good job so far.”
          “Well, it really hasn’t but … wait, what’s that red line?”
          “Oh, that’s a cubic fit model. We think that’s even BETTER than the IHME.”
          “WHAT?!”

          While a simplistic curve is useful for demonstrating the concept of “bending the curve”, you want to use more sophisticated methods for policy. The difference between Hasset’s “model” and the IHME is about 60,000 deaths. The difference between basing policy on Hassett’s model and anything else could be hundreds of thousands of deaths and millions of people getting very very sick.Report

          • Avatar Pinky in reply to Michael Siegel
            Ignored
            says:

            Thanks for the clarification.Report

          • Avatar George Turner in reply to Michael Siegel
            Ignored
            says:

            The White House tweet said

            To better visualize observed data, we also continually update a curve-fitting exercise to summarize COVID-19’s observed trajectory. Particularly with irregular data, curve fitting can improve data visualization. As shown, IHME’s mortality curves have matched the data fairly well.

            They just call it a “curve-fitting exercise” to summarize, and refer to it as helping with data visualization. There’s no indication they threw in a cubic fit for anything other than comparison to the real data.

            I suspect this outrage might be like CNN’s article yesterday that talked about the sound of a flushing toilet during oral arguments before the Supreme Court.Report

            • Avatar Michael Cain in reply to George Turner
              Ignored
              says:

              They just call it a “curve-fitting exercise” to summarize, and refer to it as helping with data visualization. There’s no indication they threw in a cubic fit for anything other than comparison to the real data.

              My complaint is that we know the underlying process is a sigmoid, or a sum of sigmoids. Most cumulative social processes look like a sigmoid — epidemics, consumer adoption of products, etc. Given that, I complain when people say, “Oh, we’ll use a cubic model here.” If they’re going to do curve-fitting, all least use a curve that matches the underlying process and extrapolates in a sane fashion, rather than a cubic that’s going to do insane things as soon as you move outside the fitted region.Report

            • Avatar Philip H in reply to George Turner
              Ignored
              says:

              Except its not. Not by a long shot.

              One of the things that really rankles us scientists is when our data are presented without the assumptions that underly them and the uncertainty that comes with them. One of the many problems here is that while the IHME’s curves are a reasonable fit visually, they aren’t as good as other models statistically, and the cubit model is even worse. This means the IHME and cubic models probably use less refined starting (or initializing conditions) and less precise assumptions then the UW model (as one example). That means that if you look at actual dosts from the model making up the curve, you see they are “noisier” and thus less precise.

              Why does precision matter? Because policies made on imprecise data mean policies aimed at the wrong goal, which in turn means policies that are less effective, and more wasteful of things like time and money. And in this case more wasteful of people’s lives.Report

              • Avatar George Turner in reply to Philip H
                Ignored
                says:

                Well, compare that to do the academic’s model in the UK.

                Code review of Fergoson’s model

                All I can say is “Yikes.”

                Why would they name a big structure P when P is obviously a name for a pointer? Any competent programmer would have called it “my_struct” and pass it to “do_function((void *)). ^_^

                There are more fundamental problems in trying to model the outbreak, because one people and governments started responding, the math became less important than the messaging, as when you’re trying to predict how many people will take the ice bucket challenge next month.

                Or, taking a page from sophisticated aerodynamic simulation and modeling, where you use well understood data and formulas for thrust, drag, lift, and weight to predict a vehicle’s longer-term future position. Sometimes the most important thing to know is the pilot’s destination. Is he going to San Diego or New York? If you don’t know the answer to that question, all the fancy math is irrelevant.Report

              • Avatar Oscar Gordon in reply to George Turner
                Ignored
                says:

                BTW Everyone should read that code review.Report

              • Avatar Michael Cain in reply to Oscar Gordon
                Ignored
                says:

                I’ll certainly concede a number of bad practices by the academics. OTOH, some of the criticism is unfair. If you replace a module doing some approximation with another one that implements a different approximation, hardcore regression testing goes out the window. Intermediate values are different, so the same inputs produce different final results. There are lots of things that should be the same — data passing through unchanged code should still produce the same results — but those test points may not be readily accessible.

                If the insurance companies were doing this, we’d still be waiting while they negotiated the specs and price. At least based on my experience, the first versions of new models are always built on the cheap. Academics are cheap but suffer from all sorts of problems — oddly trained minions graduate students for labor, outside time pressures on that labor, frequent turnover.

                It’s been that way for a very long time. I spent an afternoon at one of the Smithsonians once reading the letters that went back and forth between the professors working on discovering the neutron’s properties, chain reactions, and ultimately whether the bomb would work. I loved one of them that read roughly, “Your idea of the 5th is very interesting. However, Bob graduated and is not available to work on it. I expect a new graduate student at the beginning of the term in a couple of months, and will throw him at that problem to see if he sinks or swims.”Report

              • Avatar Oscar Gordon in reply to Michael Cain
                Ignored
                says:

                One of the code bases I am currently responsible for was written by a former academic whose approach to software development was as slipshod as the review suggests, so my sympathy for such practices is more than a little strained.

                That said, 2 points:

                1) I don’t expect academics to write clean code. They aren’t trained in it, and they rarely have the time nor the budget to do it right. They hammer out the models, and when it’s mature enough, it gets kicked out into the wild, where other people distill the core algorithms and turn it into production code.

                2) Knowing this, it is irresponsible to be basing any kind of government policy on such a code base. I know it happens, in big ways and small, every day across the world, but that does not make it a good idea.

                I mean, the whole time I was at the Lazy B*, every single modeling code had to be put through a rather rigorous test regime to validate it against known data (read: static, dynamic, wind tunnel, and flight test), and even then, all it did was reduce the number of physical tests you had to do.

                One of my coworkers has been building and maintaining a tool for the Nuke Power industry that helps Nuke engineers setup reactor heat transfer simulations, and the rigor just that tool undergoes, much less our main CFD solver, is pretty tight.

                So while I’m happy to give academics a pass on bad coding practices, I’m not going to extend the same when they shop their models around to government agencies (or media outlets) in order to influence policy. If your code has the potential to save or end lives, it better be fecking tight.

                *And don’t get me spun up again on MCAS.Report

              • Avatar Oscar Gordon in reply to Philip H
                Ignored
                says:

                One of the things that really rankles us scientists is when our data are presented without the assumptions that underly them and the uncertainty that comes with them.

                Preach it, brother!Report

  2. fillyjonk fillyjonk
    Ignored
    says:

    Figures don’t lie, but liars figure.

    Or, another old saying – which I only remember now as the back-translation-into-English a Russian fellow grad student of mine gave one day: “There are three kinds of lies. The lie, the intentional lie, and the statistic.” (I think the original – possibly Twainian – English had “damn lie” for the second category? But I like Sasha’s version of it better.)

    Ironically, about three weeks ago, I covered polynomial regression in my (now entirely online) advanced biostats class and one of the big points I made was that overfitting curves, or exuberantly fitting without considering the realities of the situation, would steer you wrong every time. But of course, I’m just a nobody at a possibly-doomed small public university….Report

    • Avatar Philip H in reply to fillyjonk
      Ignored
      says:

      Its all about assumptions and the data used for model initialization.Report

      • fillyjonk fillyjonk in reply to Philip H
        Ignored
        says:

        Yeah, and that’s the problem: with computer stats packages you have this flashy shiny tool where you don’t have to consider “what does this actually mean” or “what are the underlying assumptions” and you get bad models that suck people in because a “Real Scientist” made them.

        I’ve fallen prey to it a couple times in this, because I want Answers and I am just a stupid ecologist who doesn’t know this stuff. I’m fast learning that some of the people getting big bucks to know this stuff don’t know much more stats than *I* do and that’s kind of scary.Report

      • Avatar Michael Cain in reply to Philip H
        Ignored
        says:

        There’s model correctness as well, particularly for extrapolating (forecasting). The IHME model has the benefit of using a sane underlying model — eg, you can extrapolate indefinitely into the future and it won’t ever forecast more than 100% of the population will get sick or die. If the WH wants to do curve-fitting exercises, there’s no reason they can’t fit functions that have the same kind of sanity properties as the IHME — except that you can’t do it on Excel.Report

        • Avatar George Turner in reply to Michael Cain
          Ignored
          says:

          Excel works fine for the proper epidemic logistic curve simulations. I wrote one months ago that made day-by-day predictions of new cases, total cases, deaths, etc., though I haven’t bothered with keeping it updated.

          The trouble with the exercise is that R0 is wildly variable based on people’s behavior and prevention measures, and some of those are effective enough to drive the case numbers toward zero (such is an New Zealand). The logistic curve’s input is the final expected number of cases when the infection stops (herd immunity), which is essentially a guess based on R0.

          So you face the nagging question of whether it’s better to have a really good model of a bad guess or a really bad model of a good guess.Report

    • Avatar InMD in reply to fillyjonk
      Ignored
      says:

      Figures don’t lie, but liars figure.

      That’s a great saying I’d never heard before and I’m definitely stealing it.Report

    • Avatar Michael Cain in reply to fillyjonk
      Ignored
      says:

      Twain popularized “lies, damned lies, and statistics.” Twain attributed it to Disraeli, but it doesn’t appear in any of Disraeli’s known works. The phrase did appear in print sometime in 1891, but the author of that particular piece was anonymous.

      One of the most fun (and most difficult) math classes I took as an undergrad was approximation theory. At that time, before mammoth amounts of computer storage were available to support table look-up, it was common to have approximations to difficult-but-useful functions that had guaranteed maximum errors over particular intervals. An easy-to-evaluate rational polynomial expression might match some other function to 10 or 12 digits on a specific interval — and would blow up as soon as you moved outside that interval. It made me very cautious, perhaps overly so, about extrapolating outside the range of the data with fitted curves.Report

  3. Avatar Kristin Devine
    Ignored
    says:

    Great piece! Thanks!!Report

  4. Avatar Eduard de Jong
    Ignored
    says:

    Assuming that presenting the cubic curve as fitting the data is not done out of malice, it is clear that it’s very hard for most non-mathematicians to grasp the exponential curve.
    One of the fundamental mathematical properties of an exponential curve is that for any piece of it there always will a quadratic, cubic or higher power polynomial curve that fits that segment very closely only to deviate form it before and after the segment where it matches.

    The basic model for an epidemic is a know physical process of exponential growth, that is a process where the amount of growth depends on the size of the population. In nature there are a number of such processes. As the environment in which growth happens is n practice always finite, iny exponential growth is always followed by a decline. The decline is usually also exponential. Typically growth stops at a certain level of saturation.
    Understanding the basic mechanism behind an observed process is the basis for interpreting the date that an be measured from the process. Those measurements will then reveal the parameters of the process. For an exponential process those parameters are the growth rate and a starting value.

    For an unknown process one could try several different curves to see which fits best, and exponential should always be one of the curves tried. That was what I did thought when studying physics in the 70’s: first plot the data on logarithmic paper, and if the line looks somewhat straight, the process you have observed is exponential, so don’t bother trying any other curve!

    The nice thing about exponential growth that it has one critical parameter: growth rate. All measurable effects are proportional to that parameter: For Covid-19 the number of cases, the number deaths, the number of undiagnosed cases are all proportional to the growth rate. While it is hard to know these proportions exactly, we can chose the measurement we can measure the most accurate to learn its single parameter. In case of Covid the number of death may be the most accurately known.Report

    • Avatar Brandon Berg in reply to Eduard de Jong
      Ignored
      says:

      Note, however, that exponential growth is a poor model for COVID-19, because the growth rate changes in response to precautions taken to limit the spread. In the US, after initial exponential growth, the growth in cumulative cases has been virtually linear for a month now (suggesting a steadily falling exponential growth rate), and the growth rate in cumulative deaths has been linear for three weeks.Report

      • Avatar Michael Cain in reply to Brandon Berg
        Ignored
        says:

        In a given population, an exponential model of Covid-19 growth is fine for an early part of the cycle. Precautions affect the exponential coefficient (eg, lengthen the doubling period), but the model is still reasonable. A linear model of Covid-19 is fine for a middle part of the cycle. A log-looking model of Covid-19 is fine for the later part of the cycle. A generalized logistics curve looks exactly like that. Logistics curves were developed in order to more accurately describe the evolution of some process in a population.

        My question is why there are so many otherwise smart people ignoring the fact that we know the process will follow a logistics (or similar) curve and insist on fitting to something else. One thought, based on my experience the last time I was in graduate school, is that Excel (and in particular, Excel on Windows) has become the standard numerical platform for a lot of academics. And Excel doesn’t support fitting to less common models.Report

        • Avatar George Turner in reply to Michael Cain
          Ignored
          says:

          Well, you probably have a point that Excel doesn’t have a built-in logistics curve, so you have to build it all yourself.

          My original logistics spreadsheet, which was tuned to data from late February through the mid-March, was pretty dead on till the end of March. For March 29th it predicted 135,288 cases and we had 144,980. Then we flattened the curve and it went out the window. By April 4th my model overpredicted by a factor of 3. By April 8th it overpredicted by a factor of 10. By April 11th it overpredicted by a factor of 20.

          My original curve had 270 million total cases and 4 million deaths by this point (based on an assumed 1.5% death rate).

          The model was great. The output was garbage because we changed the world it modeled.Report

          • Avatar Michael Cain in reply to George Turner
            Ignored
            says:

            So the underlying process is non-stationary. (We certainly hoped it was non-stationary, given initial outlooks.) Now we’re into a space where we talk about how the parameters are evolving. Which gets us back to the situation where one side (the IHME for convenience) says that the process is non-stationary, stochastic, and they’re trying to forecast how various actions will affect the evolution of the model. The other side says “a cubic whose only variable is time fitted to the data makes a nice forecast about the new cases going to zero quickly.” If you ask them what the coefficients mean, there’s no answer. Because the coefficients have no relationship to the underlying process. It’s a curve-fitting exercise, with an implication that if a quadratic looks better when they do next week’s forecast, they’ll use the quadratic. That’s what I’m complaining about. If we’re just curve-fitting, let’s at least pick curves that have some relationship to the underlying process.

            Maybe I’m just frustrated. I’m working on a computer vision project and and neural networks irritate me. Thousands of coefficients to set, a reduced gradient algorithm to optimize the fit to a big pile of data, knowing that there are lots of local optima, and no one has a clue about what it means that this particular coefficient gets set to a value of 2.6. I am really hoping that this project stays simple enough that I can avoid going the NN route.Report

            • Avatar veronica d in reply to Michael Cain
              Ignored
              says:

              Yeah, I’m convinced NNs were invented to make mathematicians cry.

              One time I was talking to one of the engineers on {big NN framework}. I asked him how they do their optimization. He’s like, “Well, nothing really works well. We just do gradient descent with random restarts. It’s hit or miss. If you get a crappy result, we’ll just run it again.”

              It made me sad.

              That said, when well-tuned, NNs perform really well.

              So what does “well-tuned” mean in this case? How can I figure out how to tune it?

              “Well about that. See we can optimize over the tuning parameters using gaussian processes and…”

              Blarg!!!

              (Although gaussian processes are really cool.)

              Anyway, interpretability — what a quaint idea 🙂Report

  5. Avatar greginak
    Ignored
    says:

    I”ve long said HS kids should have at least half a year of stats even if that meant ditching some other math. Now I’m up to a year of stats with a least a quarter spent on models, there uses, misuses and challenges.Report

    • fillyjonk fillyjonk in reply to greginak
      Ignored
      says:

      Stats and probability. You can teach these without calculus. I do. I am crap at calculus but not too bad at basic stats. (Not as good at probability because I never had coursework just in that, but I have a couple books that I hope will partially fix that when I read them this summer).

      Knowing stats and how models work would go a long way to people not getting so sucked in by some of the charlatans out there – including some of the economic/goldbug/whatever-investment people.Report

      • Avatar greginak in reply to fillyjonk
        Ignored
        says:

        Same. I was barely mediocre at Calculus years ago but could do grad level stats/prob. Stats/prob/models are part of our everyday life now and we really need to know them.Report

  6. Avatar Philip H
    Ignored
    says:

    I find it frustratingly ironic that an Administration and Political Party who dismiss valid climate modeling because of statistical best fit exercises now decides pandemic response policy based on cherry picked statistical curve fitting exercises. in the grand scheme of things its probably a medium level hypocrisy but its stuff like this that makes Republican moralizing so untenable.Report

    • Avatar CJColucci in reply to Philip H
      Ignored
      says:

      “I’m shocked — SHOCKED! — that there is gambling going on here!”
      “Your winnings, Captain.”Report

    • Avatar JS in reply to Philip H
      Ignored
      says:

      “exercises now decides pandemic response policy based on cherry picked statistical curve fitting exercises”

      That is a very, very generous explanation of “Let’s toss these data points into Excel and ask it to fit them”.

      “I make pretty picture I don’t understand” is more accurate. No actual math, science, or modelling was done. Someone hit a button in Excel that they don’t understand, and assumed the magic of computers generated a model and not a simple function.Report

  7. Avatar Chip Daniels
    Ignored
    says:

    We have long ago passed the point of presumption of good faith bumbling.

    This Administration and its party has a Soviet level of contempt even for the concept of expertise. All organs of the state, all institutions of the nation, are to be bent to the service of the ruling class and all their statements and arguments are merely word screens to that end.

    We shouldn’t take anything they say seriously, since they themselves don’t.Report

  8. Avatar Chip Daniels
    Ignored
    says:

    Related from Steve M over at No More Mister Nice Blog:
    :
    DISPATCHES FROM THE AMERICAN IDIOCRACY
    https://nomoremister.blogspot.com/2020/05/dispatches-from-american-idiocracy.html

    He notes that the current number one best seller at Amazon is a book of anti-vaxxer woo describing the current pandemic as a vast conspiracy by a corrupt cabal of government and big tech, with excited blurbs from conservative amplifiers Michelle Malkin and Ben Garrison.

    Do Malkin and Garrison really believe this? Or is it a grift where they sold their name for a piece of the action?
    Who knows, and what does it matter in the end?

    Because the modern conservative movement is like an evangelical huckster organization, where belief and lies are just tools to be deployed in whatever fashion is needed at the moment to fill the collection plate.

    When it is convenient to downplay the pandemic, they shrug off as merely the flu; When it becomes important to present the pandemic as an apocalypse, they fan the flames of fear and paranoia.

    But always, always, the eye is on the grift, the fleecing of the rubes by whatever means. Whether it is to stampede the rubes into voting for tax cuts for billionaires, or getting them to agree to go back to work and drop their unemployment claims, or maybe to buy some miracle elixir hawked by a radio jock, the transfer of money upwards is the overriding goal.Report

    • Avatar Pinky in reply to Chip Daniels
      Ignored
      says:

      I hope the book isn’t what you describe, but I also think it isn’t. I mean, I don’t want to see an anti-vaxxer gain any credibility, but looking around online and checking the book’s table of contents, I don’t think it’s about covid-19 at all.Report

      • Avatar Chip Daniels in reply to Pinky
        Ignored
        says:

        The author is a certified loon:
        https://www.syracuse.com/coronavirus/2020/05/youtube-removes-plandemic-video-with-coronavirus-claims-by-dr-judy-mikovits.html
        “The Plandemic,” a 25-minute clip from an upcoming documentary, was taken off of YouTube this week for violating the Google-owned video site’s community guidelines. The video centered on Dr. Judy Mikovits, a former chronic fatigue researcher who claims the federal government is behind a “plague of corruption” to inflate profits from a potential vaccine even as COVID-19 threatens lives.

        The bigger point here is the nexus with the conservative faction which controls over half of our political system.

        It isn’t possible to speak of the conservative movement in terms of ideas and theory or conventional political ideology.
        Instead, it is just a loose confederation of grifters, charlatans, google eyed true believers and cynical special interests.

        That’s why I become so hostile to the idea of taking the President’s Tweets or the actions of Mitch McConnell and discussing them in cool cerebral terms as if they represented coherent thoughts.

        Its a bit like hearing a pronouncement of Lysenko and then discussing the pros and cons of socialist theory on corn seedlings. The entire concept is mad.Report

        • Avatar veronica d in reply to Chip Daniels
          Ignored
          says:

          It isn’t possible to speak of the conservative movement in terms of ideas and theory or conventional political ideology.
          Instead, it is just a loose confederation of grifters, charlatans, google eyed true believers and cynical special interests.

          I wonder what Jacob Wohl has been up to lately? (she asks ironically)Report

          • Avatar JS in reply to veronica d
            Ignored
            says:

            If that wasn’t sarcasm, he’s caught up in yet another false sexual harassment claim — he hired someone to accuse Fauci of it and was in the process of hiring a second, when the first one apparently just up and confessed complete with recordings of her conversation with Wohl.

            How he has not actually been arrested yet is beyond me, or at least sued.Report

            • Avatar veronica d in reply to JS
              Ignored
              says:

              It was definitely sarcasm.Report

              • Avatar JS in reply to veronica d
                Ignored
                says:

                It’s super hard to keep up with the insane news these days.

                This is the dumdest timeline.

                What’s the joke? This is the season where the directors and writers have just run out of ideas and they’re just throwing random crap at the wall? “What if Donald Trump was President? What if there was a plague? What if there was this one 20 year old guy, with backwards sunglasses, that kept trying to be a player and paying his GF to make felony accusations against people?”

                At this point, I’m expecting Ivanka Trump to announce her previously unknown cousin Willy is moving into the WH, that lovable scamp.Report

              • Avatar Oscar Gordon in reply to JS
                Ignored
                says:

                What if there was this one 20 year old guy, with backwards sunglasses, that kept trying to be a player and paying his GF to make felony accusations against people?”

                Wait? What?!Report

              • Avatar Michael Siegel in reply to Oscar Gordon
                Ignored
                says:

                I think he’s referring to Jacob Wohl, who did another of his “accuse someone famous of sexual misconduct things”, this time targeting Fauci.Report

        • Avatar Pinky in reply to Chip Daniels
          Ignored
          says:

          I think we’d both agree that a political movement is lost when its followers stop caring whether what they’re saying is true. But it took me about 10 minutes to find out that what you posted isn’t true, or at least to find good reason to believe that it isn’t true, and when I questioned you about it, you said that it doesn’t affect the bigger point. What am I supposed to do with that? I mean, if you’re right, you’re right, but I don’t sense that you care whether you’re right.

          You mention Malkin’s and Garrison’s blurbs, but you omit that Robert Kennedy Jr. wrote the forward. Anti-vaxxers show up on the left and the right, but you use this as an opportunity to denounce Mitch McConnell? I don’t think you’re doing it in bad faith, but I think you’ve badly lost perspective.Report

          • Avatar Chip Daniels in reply to Pinky
            Ignored
            says:

            First of all, we are both getting the contents of the books second hand, so bad on us.
            And yes, RFK Jr. is also a loon who should be regarded with scorn.

            Which leaves us with my point is that the conservative movement has no core beliefs, but is just a collection of grifters etc.

            Can you find grifters on the left? Absolutely.

            Do you think the two factions, liberal and conservative, are equally bereft of ideas and equally infested with grifters and charlatans?Report

Leave a Reply

Your email address will not be published. Required fields are marked *